1 line
259 KiB
JavaScript
1 line
259 KiB
JavaScript
AoPS.preload_topics = {};AoPS.preload_topics[3]={"2847354":{"num_posts":25,"posts_data":[{"post_id":25235039,"topic_id":2847354,"poster_id":702401,"post_rendered":"um how do you not silly<br>\npro tips needed<br>\nthanks<br>\nliterally sillied on at least 6 problems in mc nats and uh yeah now i'm 154th and i could've gotten a 56 if i didn't silly at all bruh","post_canonical":"um how do you not silly\npro tips needed\nthanks\nliterally sillied on at least 6 problems in mc nats and uh yeah now i'm 154th and i could've gotten a 56 if i didn't silly at all bruh\n","username":"mathmusician","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1652889294,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_702401.png?t=1645769924","num_posts":131,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2847354,"comment_count":25,"num_deleted":1,"topic_title":"how do you not silly","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25235039,"first_poster_id":702401,"first_post_time":1652889294,"first_poster_name":"mathmusician","last_post_time":1653074266,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_702401.png?t=1645769924","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_595281.png","last_post_id":25250664,"last_poster_id":595281,"last_poster_name":"heheman","last_update_time":1653074266,"category_id":3,"is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"tags":[{"tag_id":126323,"tag_text":"sillies","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"um how do you not silly<br>\npro tips needed<br>\nthanks<br>\nliterally sillied on at least 6 problems in mc nats and uh yeah now i'm 154th and i could've gotten a 56 if i didn't silly at all bruh<br>\n","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":0,"poll_id":0,"source":"","category_num_users":13,"category_num_topics":46945,"category_num_posts":659613,"num_views":466,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848400":{"num_posts":38,"posts_data":[{"post_id":25246302,"topic_id":2848400,"poster_id":560028,"post_rendered":"Thank you so much for upvoting my posts and supporting my mathcounts journey! Everyone anticipated my 3rd day updates when it got locked. So, here it is! Expect a juicy countdown scene! I appreciate critisism and discussion about my writing quality. Let's try not to get this one locked!<br>\n<br>\n1st 2 days: <a href=\"https:\/\/artofproblemsolving.com\/community\/c3h2840538\" class=\"bbcode_url\" target=\"_blank\">https:\/\/artofproblemsolving.com\/community\/c3h2840538<\/a><br>\n<br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Day 3, 5\/9\/22, JUICY COUNTDOWN SCENE<\/span><div class=\"cmty-hide-content\" style=\"display:none\">I woke up late at 7:00 because I was tired from the monuments. Energy returned when I remembered that I carried the flag! After breakfast, we lined up in the banquet hall. Each team had a purple banner with their state written in white. I was the first person to run into the countdown! It was a long wait before the run in. The director of USAMTS led me to the open door. Spotlights flashed and music blasted in the dimmed Potomac Ballroom. Already, guests and Mathcounts officials welcomed everyone. It was streaming live! I tensed, eager to burst in. This was my moment.<br>\n\u201cNow would you please stand up and join me in welcoming our Mathcounts competitors!\u201d A deep male voice boomed, \u201cAlabama!\u201d<br>\nI sprinted like it was the 60 track meet, dodging seats and following the taped arrows to our seat, leaping, hoisting my flag, yelling to the beat of thunder sticks and applause. It felt like flying. Rainbow lights beamed and cameras clicked, the lights of many stars. My teammates and I plopped in our seats, watching the other states enter. The teams were announced by a deep male voice while their name flashed in blue flames on the screen. Po shen loh sat in front of us! He turned around and waved.<br>\nFinally, the Wyoming team sat down and the team and written awards were given. Our team didn\u2019t place in the top 10. (Which would be very ironic) The top 12 individuals were:<br>\nBoya Zhang<br>\nShruti Arun<br>\nMichael Wei<br>\nEvan Fan<br>\nCalvin Wang<br>\nLiam Reddy<br>\nasdf334 Yuan! We cheered and waved our thunder sticks as Mrs. Morrison guided him to the stage. Po Shen Loh turned toward us and grinned.<br>\ndottedcaculator<br>\nVarun Gadi<br>\nRohan Garg<br>\nJohn0512 won the written competition with a score of 41!<br>\nThe 12 participants sat in seats onstage, to the right of the countdown table. It was sleek, with mathcounts logos in front of each seat. There were buzzers, a screen with the question, pencils, and paper at both of the seats. Before the round began, Julie Montoya, the moderator, asked both competitors to introduce themselves and press the buzzer. The first to answer most out of 5, but not necessarily 3, questions correctly won the round.<br>\nThe most amazing thing happened at the countdown. Although wsre, significant, and I certainly didn\u2019t qualify, we felt like the competitors onstage: the same energy, the same pressure, the same exhilaration. A long time ago, my mom said that ciphering was the most fun thing in the world. I didn\u2019t believe her then, but racing at speed, leaping over problems, is the greatest sport in the world. Now we see the 12 smartest matheletes in their prime! The beauty! The nobility! The power! The speed!<br>\nFrom the very first matchup of Evan Fan and Michael Wei, we tried to solve the problems as they were projected on screen. We were usually faster, but I don\u2019t blame them. When I used to do ciphering in 6th grade, I was faster than everyone when I wasn\u2019t onstage, but my brain froze when I was on, writing and folding the card unnaturally. It was even worse in the countdown, because there were only 3 seconds after you buzzed in to answer. Evan Fan won the first round.<br>\nWhen asdf334 went onstage, we cheered and smacked our thunder sticks. Whenever Julie asked, \u201cAre you excited?\u201d he answered, \u201cno.\u201d I understood. He had been preparing for this his whole life, and he didn\u2019t want to fail. The stakes were high. He was so fast he solved the question within 5 seconds! Julie Montoya barely read the question before he buzzed! He quickly answered 3 questions and creamed Boya Zhang.<br>\nNext, asdf334 faced dottedcaculator, who got a by. We were scared. Everyone on AoPS has predicted that dottedcaculator would win, but I didn\u2019t want that to happen. Every time asdf334 lost the question, we wished him better luck on the next question. It worked. Every time he won, we cheered with unabating enthusiasm, so powerful and energetic that Po Shen Loh whipped back and grinned at us.<br>\nStarting from the semifinals, the competitors had to answer four questions before their opponent. asdf334 gave away the answers to 2 questions to Evan Fan because answered after the 3 second limit. His voice faltered as he realized his mistake. Luckily, he answered faster in the next three questions, confident and clear. We shouted, \u201cOrz orz orz!\u201d leaped up from our chairs, and waved our thunder sticks. He would advance to the finals!<br>\nI couldn\u2019t refrain from bouncing, kicking my legs, or swinging my thunder stick. It was the final matchup: asdf334 and Calvin Wang. There was a good chance asdf334 would win, but Calvin Wang was fast. He took a long time on an easy geometry problem, so we worried. His opponent pressed the buzzer. \u201c4 root 3\u201d he answered, which was wrong! asdf334 pressed with 5 seconds remaining, \u201c12\u201d.<br>\nIt was down to the last question. \u201cA group of 9 students are split into a group of 2, a group of 3, and a group of 4. How many ways can this be done?\u201d<br>\nTime seemed to stretch, slow motion, so slow I could almost see Calvin and asdf334 calculate. I knew the method, combinations of 9 choose 2 and 7 choose 3, but I wasn\u2019t fast. Before I knew it, asdf334 slammed the red button.<br>\n\u201casdf334!\u201d Julie cried.<br>\n\u201c1260!\u201d asdf334 called.<br>\nHe was right! His name appeared in huge yellow letters on the screen.<br>\n\u201cAnd the 2022 National Mathcounts champion is- asdf334 Yuan!\u201d Julie cheered. asdf334 slammed his head against his desk, covering his face with his hands as applause and inflatable sticks thundered. He is the first champion from our state! What a victory!<br>\nHe turned around and looked at his name in disbelief. Calvin nodded, sportsmanlike.<br>\n\u201cAnd now that you have won, are you excited?\u201d Julie Montoya squealed.<br>\nasdf334 shyly shook his head. Everyone laughed, but I knew he wasn\u2019t just being humble. He got lucky. dottedcaculator, jatloe, John0512 , and Evan could\u2019ve won. For him, it wasn\u2019t exciting. It was glory and confusion and guilt. It was the weight of the championship.<br>\nEpic rock music played, a guitar melody backed by solid piano chords, grand and simply beautiful, much like the satisfaction of a well-solved problem. The melody of asdf334\u2019s victory song would touch my heart forever, because it wasn\u2019t just his victory song, it was a song for hopes and dreams, hard work and disappointment, the song I have waited to hear my entire life.<br>\nAfter he strode across the stage and received his giant glass trophy, chaos ensued. Everyone leaped up from their chairs and wrestled their way up front to glimpse at the champion. I found asdf334 at the edge of the stage, sitting in a chair with Mrs. Morrison, but the crowds were too thick. Instead, I scanned the rest of the ballroom. \u201cAre you looking for me?\u201d the most magical voice asked. I turned around to see Po Shen Loh!<br>\nI gasped, \u201cYes! asdf334 won! The first champion from our state! He is so good!\u201d<br>\nPo Shen Loh laughed, \u201cYes! You were a very supportive team. I remember you cheered for him every time he went on!\u201d<br>\nI added, \u201cWe were trying to solve the questions, but asdf334 was too fast for us. Except for the question about the inscribed circle. We were so worried!\u201d<br>\n\u201cYeah, the answer was 12 because of 30-60-90 triangles, but they hesitated. I thought, was there a trick involved?\u201d<br>\n\u201cI know! But he got it!\u201d<br>\nAfter Po Shen Loh left, I saw Richard Ruszyck in a crowd of parents. Before I reached him, I met another Mathcounts alumni. She was a MIT student and used to do mathcounts.<br>\n\u201cI don't know what I\u2019m feeling! I didn\u2019t even win, but this is the best day of my life!\u201d I cried.<br>\nShe agreed, \u201cI know, it's amazing one of your teammates won!\u201d<br>\nThen, I found Richard Ruszyck heading my way.<br>\n\u201cHi Richard!\u201d I called, \u201cThank you for creating AoPS. I couldn\u2019t have made nationals without it.\u201d<br>\n\u201cMy pleasure. Oh, you\u2019re from the Alabama team!\u201d<br>\n\u201casdf334 is the first champion from Alabama! Did you remember the year our team got 1st? I think 1991?\u201d<br>\n\u201cMathew Crawford was on the team that year! I think he placed third!\u201d<br>\n\u201cWow! Did you place?\u201d<br>\n\u201cNo. I didn\u2019t qualify for the countdown.\u201d<br>\n\u201cBut you\u2019re still so smart!\u201d<br>\n\u201cThanks. Mathew lived in Vestavia like you. He went to VHHS.\u201d<br>\n\u201cMy math teacher Mr. Taylor knows him!\u201d<br>\n\u201cTodd?\u201d It was strange hearing Mr. Taylor\u2019s first name.<br>\n\u201cYes. He teaches us geometry.\u201d<br>\n\u201cI\u2019m friends with him! Tell him I said hi!\u201d<br>\nAfter Richard left, the crowd had thinned around asdf334. I approached him, and he still sat with Mrs. Morrison. Tears streamed down his cheeks, sparkling in the spotlights. Mrs. Morrison patted his back, reminding, \u201cIt's okay.\u201d<br>\nasdf334 sniffled, \u201cI just can\u2019t- I just can\u2019t-'' His hands slid across his trophy. significant and wsre watched in disbelief. \u201cOrz orz orz.\u201d We gasped. But to me, this deserved way more than just orzing asdf334. This moment was too beautiful to pass. I kneeled down in front of asdf334\u2019s feet, balled my hands into a fist in front of me, and bent my head down until it touched the ground. I kowtowed asdf334 three times. He was perfect. But he was vulnerable.<br>\nI admire him.<br>\nWe were still gasping, crazed, when asdf334\u2019s dad came. They took some pictures together before the reporters whisked them away. significant, wsre, and I took pictures of the other orz people and got a team picture onstage. We posed behind the countdown table, hands cupped over the red buttons, ready to press. An official announced that we must leave the room in 5 minutes, so we went outside.<br>\nPo Shen Loh chat in Chinese to some parents about his math talks. I seamlessly joined. The crowd stared in surprise at my fluent Chinese as I answered some questions. Then, Po Shen Loh turned toward significant, wsre, and our parents.<br>\nWe continued to discuss the countdown questions until it was time for lunch. I ate with John0512 , dottedcaculator, Michael Wei, Rohan Garg, more countdown qualifiers and top 25% qualifiers. Finally, Mrs. Morrison and asdf334 returned. It was exciting to be surrounded by so many geniuses!<br>\nAfter lunch, ew returned to the Potomac ballroom for the Math Video Challenge. While the finalist videos played, we tried to convince each other to vote for our favorite videos. math4life2020 pulled up a chair and challenged asdf334 to an arm wrestling match, but asdf334 refused. He challenged dottedcaculator instead, and that ended in a tie. Just when math4life2020 tried to wrestle dottedcaculator, significant wrestled me over to see my results.<br>\nI did not like my results. It was not as bad as I feared, but I was the reason we got 11th place instead of 10th. I got a 23. My first chapter mathcounts score. I was so proud that I answered half of the questions then. The score itself wasn\u2019t bad, the round circle was even nice, but I could\u2019ve done better. If only I didn\u2019t silly two target questions! I would\u2019ve gotten a 27 and we would\u2019ve made top 10! I was so scared I didn\u2019t even see my rank- let\u2019s just say it was 110. Even worse than my rank, I dragged the team. I was so selfish, only caring about my enjoyment, that I forgot that my score mattered to the team!<br>\nThey chased us out of the ballroom so they could prepare for the Mathcounts party. I didn\u2019t want to see asdf334, wsre, or significant. I could only imagine the insults, accusations, and hate they would pile atop me. Instead of heading to my room like everyone else, I went to the gift shop instead to distract myself.<br>\nThe bell tinkled cheerfully when I opened the class door. The Burmese shop manager sat behind a counter. The shiny ornaments and models attracted me. I debated over whether I should get a crystal or a jeweled egg, but something made me gasp. It was a model with the major monuments on it, big and heavy enough to need two hands to carry. I ran my finger over the carved details. It was only 20 dollars, but It seemed brittle and fragile. Above the shelf, a bowl full of rhinestone jewels gleamed. I settled on two hairpins, taking advantage of the 2 for 20 deal, because they captured the sparkling spotlights and glory of the countdown perfectly. They were much easier to carry than that model.<br>\nAfter placing my purse and the gifts in my room, I returned to the lobby to find Kevin and ezpotd slumped in the chairs, scrolling on their phones. They both placed in the top 56, but they were sad, especially ezpotd. \u201cThe graders couldn\u2019t read our 6s!\u201d he complained, \u201cJust because of a handwriting error!\u201d I tried to comfort them by telling them how bad I did. I felt better, having friends to support me, and I realized that I was also the reason we didn\u2019t get 12th place or worse. And 110th was still in the top 50%.<br>\nKevin and ezpotd were USAJMO qualifiers. \u201cHow do you get good and qualify?\u201d I asked.<br>\nThey suggested I print out and practice a lot of AIME problems. They were hard and would help me be better at AMC10s.<br>\nJust when I felt better, asdf334, dottedcaculator, the New Jersey team, the Connecticut team, the California team, and the Illinois team joined us.<br>\n\u201cI talked to Po Shen Loh for 20 minutes! Are you jealous?\u201d asdf334 teased. We only chatted a little before asdf334\u2019s dad pulled asdf334 away to shop.<br>\nHalf an hour later, significant and wsre joined us, the crowd had grown to around 20 people, and we threw thunder sticks around when asdf334 returned with Sprite, Water, and a bag of Trollis. Everyone except me grabbed handfuls. I accepted a water bottle from his dad. We signed yearbooks and started For the Win games. Although the countdown was over, the rest of those who didn\u2019t make the top 12 still wanted to join the fun! Victor Chen from the Illinois team was kind enough to help me access For the Win on my phone, but we were too late to join asdf334\u2019s massive game. It was hard to play For the Win on a phone because I couldn\u2019t find fraction slashes or decimal points! In our own little games, he won most of them, but I answered some questions faster than him and won a few games.<br>\nVictor was about to teach me generating functions when Mrs. Morrison called me over.<br>\n\u201cI would like to meet you in your room.\u201d she said. Oh no. I had to give up FTW games with Victor to lead to my doom.<br>\n\u201cMy rank! My rank! My rank!\u201d I kept panicking as we went up the secret staircase.<br>\n\u201cNo. It\u2019s going to be good.\u201d She insisted.<br>\nFinally, I unlocked my room. Thankfully, peppapig was gone so she wouldn\u2019t see me cry over my rank. I froze in front of my table, silent. Peppapig had a two time competitor certificate. I had the simple participant plaque. I remember how much I admired Carol\u2019s plaque two years ago at Mrs. Li\u2019s house. How my simple plaque disgusted me!<br>\n\u201cI\u2019m so proud of you.\u201d Mrs. Morrison said, \u201cso I wanted to give you a gift. I wanted to give it here because you know, boys don\u2019t care about this.\u201d<br>\nShe revealed a large box. I lifted the lid and gasped. Nestled in wrapping paper, the exact model I wanted! I thanked her, but I also felt like I had to tell her something.<br>\n\u201cHow do you think the rest of the team would feel? I got above 20, but my rank was sub 100, 110 in fact. I am the reason we got 11th. Everyone else did better than me. They would blame me for not being in the top 10.\u201d<br>\n\u201cI know, it's tough, but you did great. Last year, we got 13th place. And you\u2019re the only girl on the team.\u201d<br>\nMrs. Morrison comforted me, but I couldn\u2019t help but feel a twinge of shame. I was proud to be a girl who does math when I win female STEM competitions or extra perks. Besides, it is an excuse when I do bad. But then again, I must prove that girls can do math as well as, and better, than boys. But I am ashamed when I am one of the few girls in the room, living proof that girls are bad at math, and the orz boys I hang out with joke about my gender.<br>\n\u201cYeah, at least I got top 50% and we didn\u2019t get 12th or worse.\u201d<br>\nAs we headed to the party, I realized, champion coaches not only coach champions; they make their entire team feel like champions.<br>\nThe escalators still weren\u2019t working, so I ran down the stairs. I grabbed glow sticks and a Mathcounts pin that looked like a medal. The open doors revealed a dimmed room, pulsing with music and rainbow lights. Giant games of skee-ball, basketball with moving hoops, battleship, chess, and checkers took place in the same room as the countdown. A long table of space themed raffle prizes stood on stage, and Mario kart and super smash bros were projected onto the countdown screens. Some people were getting fancy washable tattoos. A DJ leads a crowd on the dance floor, instructing moves and awarding the best dancers or the minigame winners with candy.<br>\nsignificant, wsre, and I ate our last hotel dinner of Mac and cheese, chicken wings, and cookies. A Mathcounts official holding medals wanted to find asdf334, but we couldn\u2019t find him. We guessed that he would play Nintendo video games, so we walked across the ballroom. Before we could make it, peppapig pulled me onto the dance floor! A Mathcounts official wearing a \u201cthis is my staff Mathcounts party shirt\u201d exclaimed, \u201cwe need more people on our team!\u201d wsre and significant fled toward the video games, where ezpotd, dottedcaculator, math4life2020, and some of my friends were already playing.<br>\nThis was my first time dancing at a party, but it wasn\u2019t as bad as I thought. I just copied the moves, which repeated as the song went on. We jammed to the macarena, \u201cgangnam style\u201d, \u201cCotton eye joe\u201d, \u201cbaby\u201d, and some songs I didn\u2019t know before a rap song blasted and the DJ started the spotlight circle. Each person had to dance, solo, in the middle for a few seconds before someone else was pushed in. peppapig pushed me in after 3 people went. I flailed and twisted for a few seconds and left. Peppapig cartwheeled onto the floor after me. After a few more rounds of dancing, I was tired. I played battleship on a giant board with pegs almost the side of soda cans. The pegs kept falling off the board when I guessed my opponent\u2019s places, so it took a long time.<br>\nBefore we even finished, they announced that it was the last song. Po Shen Loh and the AoPS crew offered to dance onstage to an easy song. Now, the dance floor was packed, the stage even more, and the DJ started \u201cParty in the USA\u201d. We didn\u2019t dance precisely in sync, but we mingled and intersected like a set of functions, individual strands tangled into a tapestry. We were of different skill levels, from different places, with different personalities and hobbies, but we were all mathletes. Although they hid behind everyone onstage, Po Shen Loh, Richard, and everyone danced a lot better than they probably thought.<br>\nToo soon, too soon, the last song ended. I talked to Po Shen Loh one last time and praised his dancing. He grinned boyishly, \u201cI just got in the groove.\u201d I nodded. This groove. Collecting pins, sightseeing, cheering at the countdown, dancing, all magical experiences packed in three days that I have been working for in the past three years.<br>\nI didn\u2019t want to leave, but the officials chased us out. After we found asdf334 outside the ballroom, we agreed to wake up the next morning at 5:00. Peppapig was already in her room, messaging on Discord. We would both leave to catch the bus at 5:30 am. That night, I tried to recount everything about the Nationals. Each day seemed long, but it ended too fast. They were the three best days of my life.<\/div><br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Day 4, 5\/10\/22, the end of a dream<\/span><div class=\"cmty-hide-content\" style=\"display:none\">\u201cBye, dragnin!\u201d peppapig called. A suitcase rolled across the carpet and the door clicked, locked. I barely even got to know my roommate and it was all over! I jumped up, packed the rest of my belongings, and raced out the door. wsre and significant were already in the lobby with Mrs. Morrison. The hotel had already prepared us breakfast: a croissant, a cinnamon bun, juice, and fruit. It was 5:20 and we waited for asdf334. Illuminated windows glowed like stars in the dark, foggy morning. Finally, asdf334 and his dad came.<br>\nWe deposited our room cards into a box. We piled into the same shuttle bus that took us here. We gazed at the sleek, mirrored windows of the Renaissance Hotel for the last time. The sky was milky marble like the hotel countertops. The bus trundled past the Tesla store, the splendid Chinatown gate, Capitol Hill, the Trump Hotel; we crossed a bridge next to the Jefferson Monument. Yellow blended at the horizon, sunbeams shattering azure cracks into marble clouds. The bus dropped us off at the Reagan airport.<br>\nAfter we checked in, Mrs. Morrison took us to Chick-Fil-A. Of course, asdf334 bought Sprite. We sat together with our suitcases, gazing over the railing. A bird inside flitted about the high ceiling arches, cheeping blithely as it explored the golden beams. The long concourse shimmered silver. Suddenly, a circular reflection rose across the glass like a spotlight. The sun bounced, minute by minute, layer by layer, upwards, gleaming like the champion\u2019s medal, radiating brilliant white streaks, beaming on the spotless airport floor, the sleek airplanes, our four, glowing faces.<br>\nI could only imagine the brightest mathletes in the Nation gazing at this very sunrise, the hope and dreams for high school olympiads, for the future, the last moment we\u2019ll share, before we all returned to our respective states.<br>\nThe sun\u2019s magical gold was replaced by blazing white by the time we boarded the plane. I sat in a window seat again, and it felt effortlessly natural when the plane took off. Maybe I was born to fly. The D.C. landscape spread beneath the cloudless sky and we, with the Washington monument piercing the sky like a heavenly pillar. The plane kept tilting as we ascended, so my view oscillated between pure sky and a diminishing landscape. After we reached cruising altitude, significant watched The Matrix while wsre and asdf334 played Sudoku.<br>\nAfter a while of trying to understand The Matrix without captions and solving asdf334\u2019s sudoku in my head, I gazed out the window. Squares of farmland all shades of green. White-roofed clusters of suburbs. Grey, grid-like cities. Then, the red-ridged Applachians, creased like carelessly scrunched napkins, but still majestic. Everyone, from asdf334, significant, and wsre to John0512, dottedcaculator, and peppapig to ezpotd, Kevin, and Michael to Po Shen Loh and Richard and everyone else I met at the nationals, all belonged to this world. And we all have a part in making tomorrow better than today.<br>\nThe flight attendants gave us, especially asdf334, one last shout out before the plane descended. Birmingham steel skyscrapers and cars scurrying on highways approached our view. The wheels grinded against concrete on the long runway, and we arrived at our gate. I carried all my belongings in 2 backpacks, so I didn\u2019t have to retrieve my valet bag. We walked in the long, sloping gateway into the airport. wsre and significant\u2019s dad met us at the baggage claim. The parents thanked Mrs. Morrison for taking care of us.<br>\n\u201cIt was a pleasure!\u201d she exclaimed, \u201cAnd now we\u2019ll have to return to normal. No more shoutouts.\u201d<br>\n\u201cNo more sightseeing.\u201d<br>\n\u201cMy school wouldn\u2019t even care.\u201d asdf334 smirked.<br>\n\u201cNo more Richard or Po Shen Loh or asdf334 for a while.\u201d I sighed.<br>\nMrs. Morrison, Katherine, wsre, his dad, and I waved goodbye to asdf334 and significant\u2019s families as they slipped out of the automatic doors. Even in the same state, we would go our separate ways. But we would always be the 2022 Alabama Nationals team, the champion\u2019s team. We would meet again at the Amsignificantan Regions Math League, a high school version of Mathcounts.<br>\nAs for now, my beloved middle school mathcounts years are over. Time to practice for the AMC10s and aim for the next stars\u2026<\/div>","post_canonical":"Thank you so much for upvoting my posts and supporting my mathcounts journey! Everyone anticipated my 3rd day updates when it got locked. So, here it is! Expect a juicy countdown scene! I appreciate critisism and discussion about my writing quality. Let's try not to get this one locked!\n\n1st 2 days: [url]https:\/\/artofproblemsolving.com\/community\/c3h2840538[\/url]\n\n[hide=Day 3, 5\/9\/22, JUICY COUNTDOWN SCENE]I woke up late at 7:00 because I was tired from the monuments. Energy returned when I remembered that I carried the flag! After breakfast, we lined up in the banquet hall. Each team had a purple banner with their state written in white. I was the first person to run into the countdown! It was a long wait before the run in. The director of USAMTS led me to the open door. Spotlights flashed and music blasted in the dimmed Potomac Ballroom. Already, guests and Mathcounts officials welcomed everyone. It was streaming live! I tensed, eager to burst in. This was my moment. \n\u201cNow would you please stand up and join me in welcoming our Mathcounts competitors!\u201d A deep male voice boomed, \u201cAlabama!\u201d\nI sprinted like it was the 60 track meet, dodging seats and following the taped arrows to our seat, leaping, hoisting my flag, yelling to the beat of thunder sticks and applause. It felt like flying. Rainbow lights beamed and cameras clicked, the lights of many stars. My teammates and I plopped in our seats, watching the other states enter. The teams were announced by a deep male voice while their name flashed in blue flames on the screen. Po shen loh sat in front of us! He turned around and waved. \nFinally, the Wyoming team sat down and the team and written awards were given. Our team didn\u2019t place in the top 10. (Which would be very ironic) The top 12 individuals were:\nBoya Zhang\nShruti Arun\nMichael Wei\nEvan Fan\nCalvin Wang\nLiam Reddy\nasdf334 Yuan! We cheered and waved our thunder sticks as Mrs. Morrison guided him to the stage. Po Shen Loh turned toward us and grinned. \ndottedcaculator\nVarun Gadi\nRohan Garg\nJohn0512 won the written competition with a score of 41! \nThe 12 participants sat in seats onstage, to the right of the countdown table. It was sleek, with mathcounts logos in front of each seat. There were buzzers, a screen with the question, pencils, and paper at both of the seats. Before the round began, Julie Montoya, the moderator, asked both competitors to introduce themselves and press the buzzer. The first to answer most out of 5, but not necessarily 3, questions correctly won the round.\nThe most amazing thing happened at the countdown. Although wsre, significant, and I certainly didn\u2019t qualify, we felt like the competitors onstage: the same energy, the same pressure, the same exhilaration. A long time ago, my mom said that ciphering was the most fun thing in the world. I didn\u2019t believe her then, but racing at speed, leaping over problems, is the greatest sport in the world. Now we see the 12 smartest matheletes in their prime! The beauty! The nobility! The power! The speed! \nFrom the very first matchup of Evan Fan and Michael Wei, we tried to solve the problems as they were projected on screen. We were usually faster, but I don\u2019t blame them. When I used to do ciphering in 6th grade, I was faster than everyone when I wasn\u2019t onstage, but my brain froze when I was on, writing and folding the card unnaturally. It was even worse in the countdown, because there were only 3 seconds after you buzzed in to answer. Evan Fan won the first round.\nWhen asdf334 went onstage, we cheered and smacked our thunder sticks. Whenever Julie asked, \u201cAre you excited?\u201d he answered, \u201cno.\u201d I understood. He had been preparing for this his whole life, and he didn\u2019t want to fail. The stakes were high. He was so fast he solved the question within 5 seconds! Julie Montoya barely read the question before he buzzed! He quickly answered 3 questions and creamed Boya Zhang. \nNext, asdf334 faced dottedcaculator, who got a by. We were scared. Everyone on AoPS has predicted that dottedcaculator would win, but I didn\u2019t want that to happen. Every time asdf334 lost the question, we wished him better luck on the next question. It worked. Every time he won, we cheered with unabating enthusiasm, so powerful and energetic that Po Shen Loh whipped back and grinned at us. \nStarting from the semifinals, the competitors had to answer four questions before their opponent. asdf334 gave away the answers to 2 questions to Evan Fan because answered after the 3 second limit. His voice faltered as he realized his mistake. Luckily, he answered faster in the next three questions, confident and clear. We shouted, \u201cOrz orz orz!\u201d leaped up from our chairs, and waved our thunder sticks. He would advance to the finals! \nI couldn\u2019t refrain from bouncing, kicking my legs, or swinging my thunder stick. It was the final matchup: asdf334 and Calvin Wang. There was a good chance asdf334 would win, but Calvin Wang was fast. He took a long time on an easy geometry problem, so we worried. His opponent pressed the buzzer. \u201c4 root 3\u201d he answered, which was wrong! asdf334 pressed with 5 seconds remaining, \u201c12\u201d. \nIt was down to the last question. \u201cA group of 9 students are split into a group of 2, a group of 3, and a group of 4. How many ways can this be done?\u201d\nTime seemed to stretch, slow motion, so slow I could almost see Calvin and asdf334 calculate. I knew the method, combinations of 9 choose 2 and 7 choose 3, but I wasn\u2019t fast. Before I knew it, asdf334 slammed the red button. \n\u201casdf334!\u201d Julie cried.\n\u201c1260!\u201d asdf334 called. \nHe was right! His name appeared in huge yellow letters on the screen. \n\u201cAnd the 2022 National Mathcounts champion is- asdf334 Yuan!\u201d Julie cheered. asdf334 slammed his head against his desk, covering his face with his hands as applause and inflatable sticks thundered. He is the first champion from our state! What a victory! \nHe turned around and looked at his name in disbelief. Calvin nodded, sportsmanlike. \n\u201cAnd now that you have won, are you excited?\u201d Julie Montoya squealed. \nasdf334 shyly shook his head. Everyone laughed, but I knew he wasn\u2019t just being humble. He got lucky. dottedcaculator, jatloe, John0512 , and Evan could\u2019ve won. For him, it wasn\u2019t exciting. It was glory and confusion and guilt. It was the weight of the championship. \nEpic rock music played, a guitar melody backed by solid piano chords, grand and simply beautiful, much like the satisfaction of a well-solved problem. The melody of asdf334\u2019s victory song would touch my heart forever, because it wasn\u2019t just his victory song, it was a song for hopes and dreams, hard work and disappointment, the song I have waited to hear my entire life.\nAfter he strode across the stage and received his giant glass trophy, chaos ensued. Everyone leaped up from their chairs and wrestled their way up front to glimpse at the champion. I found asdf334 at the edge of the stage, sitting in a chair with Mrs. Morrison, but the crowds were too thick. Instead, I scanned the rest of the ballroom. \u201cAre you looking for me?\u201d the most magical voice asked. I turned around to see Po Shen Loh! \nI gasped, \u201cYes! asdf334 won! The first champion from our state! He is so good!\u201d \nPo Shen Loh laughed, \u201cYes! You were a very supportive team. I remember you cheered for him every time he went on!\u201d \nI added, \u201cWe were trying to solve the questions, but asdf334 was too fast for us. Except for the question about the inscribed circle. We were so worried!\u201d \n\u201cYeah, the answer was 12 because of 30-60-90 triangles, but they hesitated. I thought, was there a trick involved?\u201d \n\u201cI know! But he got it!\u201d \nAfter Po Shen Loh left, I saw Richard Ruszyck in a crowd of parents. Before I reached him, I met another Mathcounts alumni. She was a MIT student and used to do mathcounts. \n\u201cI don't know what I\u2019m feeling! I didn\u2019t even win, but this is the best day of my life!\u201d I cried. \nShe agreed, \u201cI know, it's amazing one of your teammates won!\u201d \nThen, I found Richard Ruszyck heading my way. \n\u201cHi Richard!\u201d I called, \u201cThank you for creating AoPS. I couldn\u2019t have made nationals without it.\u201d\n\u201cMy pleasure. Oh, you\u2019re from the Alabama team!\u201d\n\u201casdf334 is the first champion from Alabama! Did you remember the year our team got 1st? I think 1991?\u201d \n\u201cMathew Crawford was on the team that year! I think he placed third!\u201d\n\u201cWow! Did you place?\u201d\n\u201cNo. I didn\u2019t qualify for the countdown.\u201d\n\u201cBut you\u2019re still so smart!\u201d\n\u201cThanks. Mathew lived in Vestavia like you. He went to VHHS.\u201d\n\u201cMy math teacher Mr. Taylor knows him!\u201d\n\u201cTodd?\u201d It was strange hearing Mr. Taylor\u2019s first name.\n\u201cYes. He teaches us geometry.\u201d\n\u201cI\u2019m friends with him! Tell him I said hi!\u201d\nAfter Richard left, the crowd had thinned around asdf334. I approached him, and he still sat with Mrs. Morrison. Tears streamed down his cheeks, sparkling in the spotlights. Mrs. Morrison patted his back, reminding, \u201cIt's okay.\u201d \nasdf334 sniffled, \u201cI just can\u2019t- I just can\u2019t-'' His hands slid across his trophy. significant and wsre watched in disbelief. \u201cOrz orz orz.\u201d We gasped. But to me, this deserved way more than just orzing asdf334. This moment was too beautiful to pass. I kneeled down in front of asdf334\u2019s feet, balled my hands into a fist in front of me, and bent my head down until it touched the ground. I kowtowed asdf334 three times. He was perfect. But he was vulnerable. \nI admire him. \nWe were still gasping, crazed, when asdf334\u2019s dad came. They took some pictures together before the reporters whisked them away. significant, wsre, and I took pictures of the other orz people and got a team picture onstage. We posed behind the countdown table, hands cupped over the red buttons, ready to press. An official announced that we must leave the room in 5 minutes, so we went outside. \nPo Shen Loh chat in Chinese to some parents about his math talks. I seamlessly joined. The crowd stared in surprise at my fluent Chinese as I answered some questions. Then, Po Shen Loh turned toward significant, wsre, and our parents. \nWe continued to discuss the countdown questions until it was time for lunch. I ate with John0512 , dottedcaculator, Michael Wei, Rohan Garg, more countdown qualifiers and top 25% qualifiers. Finally, Mrs. Morrison and asdf334 returned. It was exciting to be surrounded by so many geniuses!\nAfter lunch, ew returned to the Potomac ballroom for the Math Video Challenge. While the finalist videos played, we tried to convince each other to vote for our favorite videos. math4life2020 pulled up a chair and challenged asdf334 to an arm wrestling match, but asdf334 refused. He challenged dottedcaculator instead, and that ended in a tie. Just when math4life2020 tried to wrestle dottedcaculator, significant wrestled me over to see my results. \nI did not like my results. It was not as bad as I feared, but I was the reason we got 11th place instead of 10th. I got a 23. My first chapter mathcounts score. I was so proud that I answered half of the questions then. The score itself wasn\u2019t bad, the round circle was even nice, but I could\u2019ve done better. If only I didn\u2019t silly two target questions! I would\u2019ve gotten a 27 and we would\u2019ve made top 10! I was so scared I didn\u2019t even see my rank- let\u2019s just say it was 110. Even worse than my rank, I dragged the team. I was so selfish, only caring about my enjoyment, that I forgot that my score mattered to the team! \nThey chased us out of the ballroom so they could prepare for the Mathcounts party. I didn\u2019t want to see asdf334, wsre, or significant. I could only imagine the insults, accusations, and hate they would pile atop me. Instead of heading to my room like everyone else, I went to the gift shop instead to distract myself. \nThe bell tinkled cheerfully when I opened the class door. The Burmese shop manager sat behind a counter. The shiny ornaments and models attracted me. I debated over whether I should get a crystal or a jeweled egg, but something made me gasp. It was a model with the major monuments on it, big and heavy enough to need two hands to carry. I ran my finger over the carved details. It was only 20 dollars, but It seemed brittle and fragile. Above the shelf, a bowl full of rhinestone jewels gleamed. I settled on two hairpins, taking advantage of the 2 for 20 deal, because they captured the sparkling spotlights and glory of the countdown perfectly. They were much easier to carry than that model. \nAfter placing my purse and the gifts in my room, I returned to the lobby to find Kevin and ezpotd slumped in the chairs, scrolling on their phones. They both placed in the top 56, but they were sad, especially ezpotd. \u201cThe graders couldn\u2019t read our 6s!\u201d he complained, \u201cJust because of a handwriting error!\u201d I tried to comfort them by telling them how bad I did. I felt better, having friends to support me, and I realized that I was also the reason we didn\u2019t get 12th place or worse. And 110th was still in the top 50%. \nKevin and ezpotd were USAJMO qualifiers. \u201cHow do you get good and qualify?\u201d I asked.\nThey suggested I print out and practice a lot of AIME problems. They were hard and would help me be better at AMC10s. \nJust when I felt better, asdf334, dottedcaculator, the New Jersey team, the Connecticut team, the California team, and the Illinois team joined us. \n\u201cI talked to Po Shen Loh for 20 minutes! Are you jealous?\u201d asdf334 teased. We only chatted a little before asdf334\u2019s dad pulled asdf334 away to shop. \nHalf an hour later, significant and wsre joined us, the crowd had grown to around 20 people, and we threw thunder sticks around when asdf334 returned with Sprite, Water, and a bag of Trollis. Everyone except me grabbed handfuls. I accepted a water bottle from his dad. We signed yearbooks and started For the Win games. Although the countdown was over, the rest of those who didn\u2019t make the top 12 still wanted to join the fun! Victor Chen from the Illinois team was kind enough to help me access For the Win on my phone, but we were too late to join asdf334\u2019s massive game. It was hard to play For the Win on a phone because I couldn\u2019t find fraction slashes or decimal points! In our own little games, he won most of them, but I answered some questions faster than him and won a few games. \nVictor was about to teach me generating functions when Mrs. Morrison called me over. \n\u201cI would like to meet you in your room.\u201d she said. Oh no. I had to give up FTW games with Victor to lead to my doom.\n\u201cMy rank! My rank! My rank!\u201d I kept panicking as we went up the secret staircase. \n\u201cNo. It\u2019s going to be good.\u201d She insisted. \nFinally, I unlocked my room. Thankfully, peppapig was gone so she wouldn\u2019t see me cry over my rank. I froze in front of my table, silent. Peppapig had a two time competitor certificate. I had the simple participant plaque. I remember how much I admired Carol\u2019s plaque two years ago at Mrs. Li\u2019s house. How my simple plaque disgusted me!\n\u201cI\u2019m so proud of you.\u201d Mrs. Morrison said, \u201cso I wanted to give you a gift. I wanted to give it here because you know, boys don\u2019t care about this.\u201d \nShe revealed a large box. I lifted the lid and gasped. Nestled in wrapping paper, the exact model I wanted! I thanked her, but I also felt like I had to tell her something. \n\u201cHow do you think the rest of the team would feel? I got above 20, but my rank was sub 100, 110 in fact. I am the reason we got 11th. Everyone else did better than me. They would blame me for not being in the top 10.\u201d\n\u201cI know, it's tough, but you did great. Last year, we got 13th place. And you\u2019re the only girl on the team.\u201d\nMrs. Morrison comforted me, but I couldn\u2019t help but feel a twinge of shame. I was proud to be a girl who does math when I win female STEM competitions or extra perks. Besides, it is an excuse when I do bad. But then again, I must prove that girls can do math as well as, and better, than boys. But I am ashamed when I am one of the few girls in the room, living proof that girls are bad at math, and the orz boys I hang out with joke about my gender. \n\u201cYeah, at least I got top 50% and we didn\u2019t get 12th or worse.\u201d \nAs we headed to the party, I realized, champion coaches not only coach champions; they make their entire team feel like champions. \nThe escalators still weren\u2019t working, so I ran down the stairs. I grabbed glow sticks and a Mathcounts pin that looked like a medal. The open doors revealed a dimmed room, pulsing with music and rainbow lights. Giant games of skee-ball, basketball with moving hoops, battleship, chess, and checkers took place in the same room as the countdown. A long table of space themed raffle prizes stood on stage, and Mario kart and super smash bros were projected onto the countdown screens. Some people were getting fancy washable tattoos. A DJ leads a crowd on the dance floor, instructing moves and awarding the best dancers or the minigame winners with candy. \nsignificant, wsre, and I ate our last hotel dinner of Mac and cheese, chicken wings, and cookies. A Mathcounts official holding medals wanted to find asdf334, but we couldn\u2019t find him. We guessed that he would play Nintendo video games, so we walked across the ballroom. Before we could make it, peppapig pulled me onto the dance floor! A Mathcounts official wearing a \u201cthis is my staff Mathcounts party shirt\u201d exclaimed, \u201cwe need more people on our team!\u201d wsre and significant fled toward the video games, where ezpotd, dottedcaculator, math4life2020, and some of my friends were already playing. \nThis was my first time dancing at a party, but it wasn\u2019t as bad as I thought. I just copied the moves, which repeated as the song went on. We jammed to the macarena, \u201cgangnam style\u201d, \u201cCotton eye joe\u201d, \u201cbaby\u201d, and some songs I didn\u2019t know before a rap song blasted and the DJ started the spotlight circle. Each person had to dance, solo, in the middle for a few seconds before someone else was pushed in. peppapig pushed me in after 3 people went. I flailed and twisted for a few seconds and left. Peppapig cartwheeled onto the floor after me. After a few more rounds of dancing, I was tired. I played battleship on a giant board with pegs almost the side of soda cans. The pegs kept falling off the board when I guessed my opponent\u2019s places, so it took a long time. \nBefore we even finished, they announced that it was the last song. Po Shen Loh and the AoPS crew offered to dance onstage to an easy song. Now, the dance floor was packed, the stage even more, and the DJ started \u201cParty in the USA\u201d. We didn\u2019t dance precisely in sync, but we mingled and intersected like a set of functions, individual strands tangled into a tapestry. We were of different skill levels, from different places, with different personalities and hobbies, but we were all mathletes. Although they hid behind everyone onstage, Po Shen Loh, Richard, and everyone danced a lot better than they probably thought.\nToo soon, too soon, the last song ended. I talked to Po Shen Loh one last time and praised his dancing. He grinned boyishly, \u201cI just got in the groove.\u201d I nodded. This groove. Collecting pins, sightseeing, cheering at the countdown, dancing, all magical experiences packed in three days that I have been working for in the past three years. \nI didn\u2019t want to leave, but the officials chased us out. After we found asdf334 outside the ballroom, we agreed to wake up the next morning at 5:00. Peppapig was already in her room, messaging on Discord. We would both leave to catch the bus at 5:30 am. That night, I tried to recount everything about the Nationals. Each day seemed long, but it ended too fast. They were the three best days of my life. \n[\/hide]\n[hide= Day 4, 5\/10\/22, the end of a dream]\u201cBye, dragnin!\u201d peppapig called. A suitcase rolled across the carpet and the door clicked, locked. I barely even got to know my roommate and it was all over! I jumped up, packed the rest of my belongings, and raced out the door. wsre and significant were already in the lobby with Mrs. Morrison. The hotel had already prepared us breakfast: a croissant, a cinnamon bun, juice, and fruit. It was 5:20 and we waited for asdf334. Illuminated windows glowed like stars in the dark, foggy morning. Finally, asdf334 and his dad came. \nWe deposited our room cards into a box. We piled into the same shuttle bus that took us here. We gazed at the sleek, mirrored windows of the Renaissance Hotel for the last time. The sky was milky marble like the hotel countertops. The bus trundled past the Tesla store, the splendid Chinatown gate, Capitol Hill, the Trump Hotel; we crossed a bridge next to the Jefferson Monument. Yellow blended at the horizon, sunbeams shattering azure cracks into marble clouds. The bus dropped us off at the Reagan airport. \nAfter we checked in, Mrs. Morrison took us to Chick-Fil-A. Of course, asdf334 bought Sprite. We sat together with our suitcases, gazing over the railing. A bird inside flitted about the high ceiling arches, cheeping blithely as it explored the golden beams. The long concourse shimmered silver. Suddenly, a circular reflection rose across the glass like a spotlight. The sun bounced, minute by minute, layer by layer, upwards, gleaming like the champion\u2019s medal, radiating brilliant white streaks, beaming on the spotless airport floor, the sleek airplanes, our four, glowing faces. \nI could only imagine the brightest mathletes in the Nation gazing at this very sunrise, the hope and dreams for high school olympiads, for the future, the last moment we\u2019ll share, before we all returned to our respective states. \nThe sun\u2019s magical gold was replaced by blazing white by the time we boarded the plane. I sat in a window seat again, and it felt effortlessly natural when the plane took off. Maybe I was born to fly. The D.C. landscape spread beneath the cloudless sky and we, with the Washington monument piercing the sky like a heavenly pillar. The plane kept tilting as we ascended, so my view oscillated between pure sky and a diminishing landscape. After we reached cruising altitude, significant watched The Matrix while wsre and asdf334 played Sudoku. \nAfter a while of trying to understand The Matrix without captions and solving asdf334\u2019s sudoku in my head, I gazed out the window. Squares of farmland all shades of green. White-roofed clusters of suburbs. Grey, grid-like cities. Then, the red-ridged Applachians, creased like carelessly scrunched napkins, but still majestic. Everyone, from asdf334, significant, and wsre to John0512, dottedcaculator, and peppapig to ezpotd, Kevin, and Michael to Po Shen Loh and Richard and everyone else I met at the nationals, all belonged to this world. And we all have a part in making tomorrow better than today. \nThe flight attendants gave us, especially asdf334, one last shout out before the plane descended. Birmingham steel skyscrapers and cars scurrying on highways approached our view. The wheels grinded against concrete on the long runway, and we arrived at our gate. I carried all my belongings in 2 backpacks, so I didn\u2019t have to retrieve my valet bag. We walked in the long, sloping gateway into the airport. wsre and significant\u2019s dad met us at the baggage claim. The parents thanked Mrs. Morrison for taking care of us. \n\u201cIt was a pleasure!\u201d she exclaimed, \u201cAnd now we\u2019ll have to return to normal. No more shoutouts.\u201d \n\u201cNo more sightseeing.\u201d \n\u201cMy school wouldn\u2019t even care.\u201d asdf334 smirked. \n\u201cNo more Richard or Po Shen Loh or asdf334 for a while.\u201d I sighed. \nMrs. Morrison, Katherine, wsre, his dad, and I waved goodbye to asdf334 and significant\u2019s families as they slipped out of the automatic doors. Even in the same state, we would go our separate ways. But we would always be the 2022 Alabama Nationals team, the champion\u2019s team. We would meet again at the Amsignificantan Regions Math League, a high school version of Mathcounts. \nAs for now, my beloved middle school mathcounts years are over. Time to practice for the AMC10s and aim for the next stars\u2026\n[\/hide]","username":"dragnin","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":37,"nothanks_received":0,"thankers":"player01, QQMath, eagles2018, samrocksnature, Coco7, asimov, michaelwenquan, UnknownMonkey, hh99754539, math31415926535, CT17, Rock22, asdf334, justJen, centslordm, v4913, doulai1, stuffedmath, AlphaBetaGammaOmega, hurdler, mahaler, Amkan2022, Jiseop55406, purplepenguin2, megarnie, peelybonehead, Bedwarspro, russellk, mathking999, bballlegend, tigeryong, bluelinfish, Significant, cj13609517288, thodupunuri, Jndd, hellomathworks10","deleted":false,"post_number":1,"post_time":1653006011,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_560028.png","num_posts":173,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848400,"comment_count":38,"num_deleted":2,"topic_title":"Dragnin's COMPLETE Mathcounts Nationals story!","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25246302,"first_poster_id":560028,"first_post_time":1653006011,"first_poster_name":"dragnin","last_post_time":1653074217,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_560028.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_931141.png","last_post_id":25250659,"last_poster_id":931141,"last_poster_name":"Swaid","last_update_time":1653074217,"category_id":3,"is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"tags":[{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true},{"tag_id":1499442,"tag_text":"historic document","is_visible":true},{"tag_id":851011,"tag_text":"math story","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Thank you so much for upvoting my posts and supporting my mathcounts journey! Everyone anticipated my 3rd day updates when it got locked. So, here it is! Expect a juicy countdown scene! I appreciate critisism and discussion about my writing quality. Let's try not to get this one locked!<br>\n<br>\n1st 2 days: <a href=\"https:\/\/artofproblemsolving.com\/community\/c3h2840538\" class=\"bbcode_url\">https:\/\/artofproblemsolving.com\/community\/c3h2840538<\/a><br>\n<br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Day 3, 5\/9\/22, JUICY COUNTDOWN SCENE<\/span><div class=\"cmty-hide-content\" style=\"display:none\">I woke up late at 7:00 because I was tired from the monuments. Energy returned when I remembered that I carried the flag! After breakfast, we lined up in the banquet hall. Each team had a purple banner with their state written in white. I was the first person to run into the countdown! It was a long wait before the run in. The director of USAMTS led me to the open door. Spotlights flashed and music blasted in the dimmed Potomac Ballroom. Already, guests and Mathcounts officials welcomed everyone. It was streaming live! I tensed, eager to burst in. This was my moment.<br>\n\u201cNow would you please stand up and join me in welcoming our Mathcounts competitors!\u201d A deep male voice boomed, \u201cAlabama!\u201d<br>\nI sprinted like it was the 60 track meet, dodging seats and following the taped arrows to our seat, leaping, hoisting my flag, yelling to the beat of thunder sticks and applause. It felt like flying. Rainbow lights beamed and cameras clicked, the lights of many stars. My teammates and I plopped in our seats, watching the other states enter. The teams were announced by a deep male voice while their name flashed in blue flames on the screen. Po shen loh sat in front of us! He turned around and waved.<br>\nFinally, the Wyoming team sat down and the team and written awards were given. Our team didn\u2019t place in the top 10. (Which would be very ironic) The top 12 individuals were:<br>\nBoya Zhang<br>\nShruti Arun<br>\nMichael Wei<br>\nEvan Fan<br>\nCalvin Wang<br>\nLiam Reddy<br>\nasdf334 Yuan! We cheered and waved our thunder sticks as Mrs. Morrison guided him to the stage. Po Shen Loh turned toward us and grinned.<br>\ndottedcaculator<br>\nVarun Gadi<br>\nRohan Garg<br>\nJohn0512 won the written competition with a score of 41!<br>\nThe 12 participants sat in seats onstage, to the right of the countdown table. It was sleek, with mathcounts logos in front of each seat. There were buzzers, a screen with the question, pencils, and paper at both of the seats. Before the round began, Julie Montoya, the moderator, asked both competitors to introduce themselves and press the buzzer. The first to answer most out of 5, but not necessarily 3, questions correctly won the round.<br>\nThe most amazing thing happened at the countdown. Although wsre, significant, and I certainly didn\u2019t qualify, we felt like the competitors onstage: the same energy, the same pressure, the same exhilaration. A long time ago, my mom said that ciphering was the most fun thing in the world. I didn\u2019t believe her then, but racing at speed, leaping over problems, is the greatest sport in the world. Now we see the 12 smartest matheletes in their prime! The beauty! The nobility! The power! The speed!<br>\nFrom the very first matchup of Evan Fan and Michael Wei, we tried to solve the problems as they were projected on screen. We were usually faster, but I don\u2019t blame them. When I used to do ciphering in 6th grade, I was faster than everyone when I wasn\u2019t onstage, but my brain froze when I was on, writing and folding the card unnaturally. It was even worse in the countdown, because there were only 3 seconds after you buzzed in to answer. Evan Fan won the first round.<br>\nWhen asdf334 went onstage, we cheered and smacked our thunder sticks. Whenever Julie asked, \u201cAre you excited?\u201d he answered, \u201cno.\u201d I understood. He had been preparing for this his whole life, and he didn\u2019t want to fail. The stakes were high. He was so fast he solved the question within 5 seconds! Julie Montoya barely read the question before he buzzed! He quickly answered 3 questions and creamed Boya Zhang.<br>\nNext, asdf334 faced dottedcaculator, who got a by. We were scared. Everyone on AoPS has predicted that dottedcaculator would win, but I didn\u2019t want that to happen. Every time asdf334 lost the question, we wished him better luck on the next question. It worked. Every time he won, we cheered with unabating enthusiasm, so powerful and energetic that Po Shen Loh whipped back and grinned at us.<br>\nStarting from the semifinals, the competitors had to answer four questions before their opponent. asdf334 gave away the answers to 2 questions to Evan Fan because answered after the 3 second limit. His voice faltered as he realized his mistake. Luckily, he answered faster in the next three questions, confident and clear. We shouted, \u201cOrz orz orz!\u201d leaped up from our chairs, and waved our thunder sticks. He would advance to the finals!<br>\nI couldn\u2019t refrain from bouncing, kicking my legs, or swinging my thunder stick. It was the final matchup: asdf334 and Calvin Wang. There was a good chance asdf334 would win, but Calvin Wang was fast. He took a long time on an easy geometry problem, so we worried. His opponent pressed the buzzer. \u201c4 root 3\u201d he answered, which was wrong! asdf334 pressed with 5 seconds remaining, \u201c12\u201d.<br>\nIt was down to the last question. \u201cA group of 9 students are split into a group of 2, a group of 3, and a group of 4. How many ways can this be done?\u201d<br>\nTime seemed to stretch, slow motion, so slow I could almost see Calvin and asdf334 calculate. I knew the method, combinations of 9 choose 2 and 7 choose 3, but I wasn\u2019t fast. Before I knew it, asdf334 slammed the red button.<br>\n\u201casdf334!\u201d Julie cried.<br>\n\u201c1260!\u201d asdf334 called.<br>\nHe was right! His name appeared in huge yellow letters on the screen.<br>\n\u201cAnd the 2022 National Mathcounts champion is- asdf334 Yuan!\u201d Julie cheered. asdf334 slammed his head against his desk, covering his face with his hands as applause and inflatable sticks thundered. He is the first champion from our state! What a victory!<br>\nHe turned around and looked at his name in disbelief. Calvin nodded, sportsmanlike.<br>\n\u201cAnd now that you have won, are you excited?\u201d Julie Montoya squealed.<br>\nasdf334 shyly shook his head. Everyone laughed, but I knew he wasn\u2019t just being humble. He got lucky. dottedcaculator, jatloe, John0512 , and Evan could\u2019ve won. For him, it wasn\u2019t exciting. It was glory and confusion and guilt. It was the weight of the championship.<br>\nEpic rock music played, a guitar melody backed by solid piano chords, grand and simply beautiful, much like the satisfaction of a well-solved problem. The melody of asdf334\u2019s victory song would touch my heart forever, because it wasn\u2019t just his victory song, it was a song for hopes and dreams, hard work and disappointment, the song I have waited to hear my entire life.<br>\nAfter he strode across the stage and received his giant glass trophy, chaos ensued. Everyone leaped up from their chairs and wrestled their way up front to glimpse at the champion. I found asdf334 at the edge of the stage, sitting in a chair with Mrs. Morrison, but the crowds were too thick. Instead, I scanned the rest of the ballroom. \u201cAre you looking for me?\u201d the most magical voice asked. I turned around to see Po Shen Loh!<br>\nI gasped, \u201cYes! asdf334 won! The first champion from our state! He is so good!\u201d<br>\nPo Shen Loh laughed, \u201cYes! You were a very supportive team. I remember you cheered for him every time he went on!\u201d<br>\nI added, \u201cWe were trying to solve the questions, but asdf334 was too fast for us. Except for the question about the inscribed circle. We were so worried!\u201d<br>\n\u201cYeah, the answer was 12 because of 30-60-90 triangles, but they hesitated. I thought, was there a trick involved?\u201d<br>\n\u201cI know! But he got it!\u201d<br>\nAfter Po Shen Loh left, I saw Richard Ruszyck in a crowd of parents. Before I reached him, I met another Mathcounts alumni. She was a MIT student and used to do mathcounts.<br>\n\u201cI don't know what I\u2019m feeling! I didn\u2019t even win, but this is the best day of my life!\u201d I cried.<br>\nShe agreed, \u201cI know, it's amazing one of your teammates won!\u201d<br>\nThen, I found Richard Ruszyck heading my way.<br>\n\u201cHi Richard!\u201d I called, \u201cThank you for creating AoPS. I couldn\u2019t have made nationals without it.\u201d<br>\n\u201cMy pleasure. Oh, you\u2019re from the Alabama team!\u201d<br>\n\u201casdf334 is the first champion from Alabama! Did you remember the year our team got 1st? I think 1991?\u201d<br>\n\u201cMathew Crawford was on the team that year! I think he placed third!\u201d<br>\n\u201cWow! Did you place?\u201d<br>\n\u201cNo. I didn\u2019t qualify for the countdown.\u201d<br>\n\u201cBut you\u2019re still so smart!\u201d<br>\n\u201cThanks. Mathew lived in Vestavia like you. He went to VHHS.\u201d<br>\n\u201cMy math teacher Mr. Taylor knows him!\u201d<br>\n\u201cTodd?\u201d It was strange hearing Mr. Taylor\u2019s first name.<br>\n\u201cYes. He teaches us geometry.\u201d<br>\n\u201cI\u2019m friends with him! Tell him I said hi!\u201d<br>\nAfter Richard left, the crowd had thinned around asdf334. I approached him, and he still sat with Mrs. Morrison. Tears streamed down his cheeks, sparkling in the spotlights. Mrs. Morrison patted his back, reminding, \u201cIt's okay.\u201d<br>\nasdf334 sniffled, \u201cI just can\u2019t- I just can\u2019t-'' His hands slid across his trophy. significant and wsre watched in disbelief. \u201cOrz orz orz.\u201d We gasped. But to me, this deserved way more than just orzing asdf334. This moment was too beautiful to pass. I kneeled down in front of asdf334\u2019s feet, balled my hands into a fist in front of me, and bent my head down until it touched the ground. I kowtowed asdf334 three times. He was perfect. But he was vulnerable.<br>\nI admire him.<br>\nWe were still gasping, crazed, when asdf334\u2019s dad came. They took some pictures together before the reporters whisked them away. significant, wsre, and I took pictures of the other orz people and got a team picture onstage. We posed behind the countdown table, hands cupped over the red buttons, ready to press. An official announced that we must leave the room in 5 minutes, so we went outside.<br>\nPo Shen Loh chat in Chinese to some parents about his math talks. I seamlessly joined. The crowd stared in surprise at my fluent Chinese as I answered some questions. Then, Po Shen Loh turned toward significant, wsre, and our parents.<br>\nWe continued to discuss the countdown questions until it was time for lunch. I ate with John0512 , dottedcaculator, Michael Wei, Rohan Garg, more countdown qualifiers and top 25% qualifiers. Finally, Mrs. Morrison and asdf334 returned. It was exciting to be surrounded by so many geniuses!<br>\nAfter lunch, ew returned to the Potomac ballroom for the Math Video Challenge. While the finalist videos played, we tried to convince each other to vote for our favorite videos. math4life2020 pulled up a chair and challenged asdf334 to an arm wrestling match, but asdf334 refused. He challenged dottedcaculator instead, and that ended in a tie. Just when math4life2020 tried to wrestle dottedcaculator, significant wrestled me over to see my results.<br>\nI did not like my results. It was not as bad as I feared, but I was the reason we got 11th place instead of 10th. I got a 23. My first chapter mathcounts score. I was so proud that I answered half of the questions then. The score itself wasn\u2019t bad, the round circle was even nice, but I could\u2019ve done better. If only I didn\u2019t silly two target questions! I would\u2019ve gotten a 27 and we would\u2019ve made top 10! I was so scared I didn\u2019t even see my rank- let\u2019s just say it was 110. Even worse than my rank, I dragged the team. I was so selfish, only caring about my enjoyment, that I forgot that my score mattered to the team!<br>\nThey chased us out of the ballroom so they could prepare for the Mathcounts party. I didn\u2019t want to see asdf334, wsre, or significant. I could only imagine the insults, accusations, and hate they would pile atop me. Instead of heading to my room like everyone else, I went to the gift shop instead to distract myself.<br>\nThe bell tinkled cheerfully when I opened the class door. The Burmese shop manager sat behind a counter. The shiny ornaments and models attracted me. I debated over whether I should get a crystal or a jeweled egg, but something made me gasp. It was a model with the major monuments on it, big and heavy enough to need two hands to carry. I ran my finger over the carved details. It was only 20 dollars, but It seemed brittle and fragile. Above the shelf, a bowl full of rhinestone jewels gleamed. I settled on two hairpins, taking advantage of the 2 for 20 deal, because they captured the sparkling spotlights and glory of the countdown perfectly. They were much easier to carry than that model.<br>\nAfter placing my purse and the gifts in my room, I returned to the lobby to find Kevin and ezpotd slumped in the chairs, scrolling on their phones. They both placed in the top 56, but they were sad, especially ezpotd. \u201cThe graders couldn\u2019t read our 6s!\u201d he complained, \u201cJust because of a handwriting error!\u201d I tried to comfort them by telling them how bad I did. I felt better, having friends to support me, and I realized that I was also the reason we didn\u2019t get 12th place or worse. And 110th was still in the top 50%.<br>\nKevin and ezpotd were USAJMO qualifiers. \u201cHow do you get good and qualify?\u201d I asked.<br>\nThey suggested I print out and practice a lot of AIME problems. They were hard and would help me be better at AMC10s.<br>\nJust when I felt better, asdf334, dottedcaculator, the New Jersey team, the Connecticut team, the California team, and the Illinois team joined us.<br>\n\u201cI talked to Po Shen Loh for 20 minutes! Are you jealous?\u201d asdf334 teased. We only chatted a little before asdf334\u2019s dad pulled asdf334 away to shop.<br>\nHalf an hour later, significant and wsre joined us, the crowd had grown to around 20 people, and we threw thunder sticks around when asdf334 returned with Sprite, Water, and a bag of Trollis. Everyone except me grabbed handfuls. I accepted a water bottle from his dad. We signed yearbooks and started For the Win games. Although the countdown was over, the rest of those who didn\u2019t make the top 12 still wanted to join the fun! Victor Chen from the Illinois team was kind enough to help me access For the Win on my phone, but we were too late to join asdf334\u2019s massive game. It was hard to play For the Win on a phone because I couldn\u2019t find fraction slashes or decimal points! In our own little games, he won most of them, but I answered some questions faster than him and won a few games.<br>\nVictor was about to teach me generating functions when Mrs. Morrison called me over.<br>\n\u201cI would like to meet you in your room.\u201d she said. Oh no. I had to give up FTW games with Victor to lead to my doom.<br>\n\u201cMy rank! My rank! My rank!\u201d I kept panicking as we went up the secret staircase.<br>\n\u201cNo. It\u2019s going to be good.\u201d She insisted.<br>\nFinally, I unlocked my room. Thankfully, peppapig was gone so she wouldn\u2019t see me cry over my rank. I froze in front of my table, silent. Peppapig had a two time competitor certificate. I had the simple participant plaque. I remember how much I admired Carol\u2019s plaque two years ago at Mrs. Li\u2019s house. How my simple plaque disgusted me!<br>\n\u201cI\u2019m so proud of you.\u201d Mrs. Morrison said, \u201cso I wanted to give you a gift. I wanted to give it here because you know, boys don\u2019t care about this.\u201d<br>\nShe revealed a large box. I lifted the lid and gasped. Nestled in wrapping paper, the exact model I wanted! I thanked her, but I also felt like I had to tell her something.<br>\n\u201cHow do you think the rest of the team would feel? I got above 20, but my rank was sub 100, 110 in fact. I am the reason we got 11th. Everyone else did better than me. They would blame me for not being in the top 10.\u201d<br>\n\u201cI know, it's tough, but you did great. Last year, we got 13th place. And you\u2019re the only girl on the team.\u201d<br>\nMrs. Morrison comforted me, but I couldn\u2019t help but feel a twinge of shame. I was proud to be a girl who does math when I win female STEM competitions or extra perks. Besides, it is an excuse when I do bad. But then again, I must prove that girls can do math as well as, and better, than boys. But I am ashamed when I am one of the few girls in the room, living proof that girls are bad at math, and the orz boys I hang out with joke about my gender.<br>\n\u201cYeah, at least I got top 50% and we didn\u2019t get 12th or worse.\u201d<br>\nAs we headed to the party, I realized, champion coaches not only coach champions; they make their entire team feel like champions.<br>\nThe escalators still weren\u2019t working, so I ran down the stairs. I grabbed glow sticks and a Mathcounts pin that looked like a medal. The open doors revealed a dimmed room, pulsing with music and rainbow lights. Giant games of skee-ball, basketball with moving hoops, battleship, chess, and checkers took place in the same room as the countdown. A long table of space themed raffle prizes stood on stage, and Mario kart and super smash bros were projected onto the countdown screens. Some people were getting fancy washable tattoos. A DJ leads a crowd on the dance floor, instructing moves and awarding the best dancers or the minigame winners with candy.<br>\nsignificant, wsre, and I ate our last hotel dinner of Mac and cheese, chicken wings, and cookies. A Mathcounts official holding medals wanted to find asdf334, but we couldn\u2019t find him. We guessed that he would play Nintendo video games, so we walked across the ballroom. Before we could make it, peppapig pulled me onto the dance floor! A Mathcounts official wearing a \u201cthis is my staff Mathcounts party shirt\u201d exclaimed, \u201cwe need more people on our team!\u201d wsre and significant fled toward the video games, where ezpotd, dottedcaculator, math4life2020, and some of my friends were already playing.<br>\nThis was my first time dancing at a party, but it wasn\u2019t as bad as I thought. I just copied the moves, which repeated as the song went on. We jammed to the macarena, \u201cgangnam style\u201d, \u201cCotton eye joe\u201d, \u201cbaby\u201d, and some songs I didn\u2019t know before a rap song blasted and the DJ started the spotlight circle. Each person had to dance, solo, in the middle for a few seconds before someone else was pushed in. peppapig pushed me in after 3 people went. I flailed and twisted for a few seconds and left. Peppapig cartwheeled onto the floor after me. After a few more rounds of dancing, I was tired. I played battleship on a giant board with pegs almost the side of soda cans. The pegs kept falling off the board when I guessed my opponent\u2019s places, so it took a long time.<br>\nBefore we even finished, they announced that it was the last song. Po Shen Loh and the AoPS crew offered to dance onstage to an easy song. Now, the dance floor was packed, the stage even more, and the DJ started \u201cParty in the USA\u201d. We didn\u2019t dance precisely in sync, but we mingled and intersected like a set of functions, individual strands tangled into a tapestry. We were of different skill levels, from different places, with different personalities and hobbies, but we were all mathletes. Although they hid behind everyone onstage, Po Shen Loh, Richard, and everyone danced a lot better than they probably thought.<br>\nToo soon, too soon, the last song ended. I talked to Po Shen Loh one last time and praised his dancing. He grinned boyishly, \u201cI just got in the groove.\u201d I nodded. This groove. Collecting pins, sightseeing, cheering at the countdown, dancing, all magical experiences packed in three days that I have been working for in the past three years.<br>\nI didn\u2019t want to leave, but the officials chased us out. After we found asdf334 outside the ballroom, we agreed to wake up the next morning at 5:00. Peppapig was already in her room, messaging on Discord. We would both leave to catch the bus at 5:30 am. That night, I tried to recount everything about the Nationals. Each day seemed long, but it ended too fast. They were the three best days of my life.<\/div><br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Day 4, 5\/10\/22, the end of a dream<\/span><div class=\"cmty-hide-content\" style=\"display:none\">\u201cBye, dragnin!\u201d peppapig called. A suitcase rolled across the carpet and the door clicked, locked. I barely even got to know my roommate and it was all over! I jumped up, packed the rest of my belongings, and raced out the door. wsre and significant were already in the lobby with Mrs. Morrison. The hotel had already prepared us breakfast: a croissant, a cinnamon bun, juice, and fruit. It was 5:20 and we waited for asdf334. Illuminated windows glowed like stars in the dark, foggy morning. Finally, asdf334 and his dad came.<br>\nWe deposited our room cards into a box. We piled into the same shuttle bus that took us here. We gazed at the sleek, mirrored windows of the Renaissance Hotel for the last time. The sky was milky marble like the hotel countertops. The bus trundled past the Tesla store, the splendid Chinatown gate, Capitol Hill, the Trump Hotel; we crossed a bridge next to the Jefferson Monument. Yellow blended at the horizon, sunbeams shattering azure cracks into marble clouds. The bus dropped us off at the Reagan airport.<br>\nAfter we checked in, Mrs. Morrison took us to Chick-Fil-A. Of course, asdf334 bought Sprite. We sat together with our suitcases, gazing over the railing. A bird inside flitted about the high ceiling arches, cheeping blithely as it explored the golden beams. The long concourse shimmered silver. Suddenly, a circular reflection rose across the glass like a spotlight. The sun bounced, minute by minute, layer by layer, upwards, gleaming like the champion\u2019s medal, radiating brilliant white streaks, beaming on the spotless airport floor, the sleek airplanes, our four, glowing faces.<br>\nI could only imagine the brightest mathletes in the Nation gazing at this very sunrise, the hope and dreams for high school olympiads, for the future, the last moment we\u2019ll share, before we all returned to our respective states.<br>\nThe sun\u2019s magical gold was replaced by blazing white by the time we boarded the plane. I sat in a window seat again, and it felt effortlessly natural when the plane took off. Maybe I was born to fly. The D.C. landscape spread beneath the cloudless sky and we, with the Washington monument piercing the sky like a heavenly pillar. The plane kept tilting as we ascended, so my view oscillated between pure sky and a diminishing landscape. After we reached cruising altitude, significant watched The Matrix while wsre and asdf334 played Sudoku.<br>\nAfter a while of trying to understand The Matrix without captions and solving asdf334\u2019s sudoku in my head, I gazed out the window. Squares of farmland all shades of green. White-roofed clusters of suburbs. Grey, grid-like cities. Then, the red-ridged Applachians, creased like carelessly scrunched napkins, but still majestic. Everyone, from asdf334, significant, and wsre to John0512, dottedcaculator, and peppapig to ezpotd, Kevin, and Michael to Po Shen Loh and Richard and everyone else I met at the nationals, all belonged to this world. And we all have a part in making tomorrow better than today.<br>\nThe flight attendants gave us, especially asdf334, one last shout out before the plane descended. Birmingham steel skyscrapers and cars scurrying on highways approached our view. The wheels grinded against concrete on the long runway, and we arrived at our gate. I carried all my belongings in 2 backpacks, so I didn\u2019t have to retrieve my valet bag. We walked in the long, sloping gateway into the airport. wsre and significant\u2019s dad met us at the baggage claim. The parents thanked Mrs. Morrison for taking care of us.<br>\n\u201cIt was a pleasure!\u201d she exclaimed, \u201cAnd now we\u2019ll have to return to normal. No more shoutouts.\u201d<br>\n\u201cNo more sightseeing.\u201d<br>\n\u201cMy school wouldn\u2019t even care.\u201d asdf334 smirked.<br>\n\u201cNo more Richard or Po Shen Loh or asdf334 for a while.\u201d I sighed.<br>\nMrs. Morrison, Katherine, wsre, his dad, and I waved goodbye to asdf334 and significant\u2019s families as they slipped out of the automatic doors. Even in the same state, we would go our separate ways. But we would always be the 2022 Alabama Nationals team, the champion\u2019s team. We would meet again at the Amsignificantan Regions Math League, a high school version of Mathcounts.<br>\nAs for now, my beloved middle school mathcounts years are over. Time to practice for the AMC10s and aim for the next stars\u2026<\/div>","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":0,"poll_id":0,"source":"","category_num_users":13,"category_num_topics":46945,"category_num_posts":659613,"num_views":946,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848745":{"num_posts":35,"posts_data":[{"post_id":25249944,"topic_id":2848745,"poster_id":839899,"post_rendered":"I live in texas. how hard is it to get to mathcounts nats","post_canonical":"I live in texas. how hard is it to get to mathcounts nats","username":"hellomathworks10","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":2,"nothanks_received":0,"thankers":"fluff_E, ImSh95","deleted":false,"post_number":1,"post_time":1653065508,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_839899.jpeg?t=1652990130","num_posts":73,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848745,"comment_count":35,"num_deleted":1,"topic_title":"mathcounts texas","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25249944,"first_poster_id":839899,"first_post_time":1653065508,"first_poster_name":"hellomathworks10","last_post_time":1653074048,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_839899.jpeg?t=1652990130","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_595281.png","last_post_id":25250645,"last_poster_id":595281,"last_poster_name":"heheman","last_update_time":1653074048,"category_id":3,"is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"tags":[{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"I live in texas. how hard is it to get to mathcounts nats","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":0,"poll_id":0,"source":"","category_num_users":13,"category_num_topics":46945,"category_num_posts":659613,"num_views":294,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2756545":{"num_posts":238,"posts_data":[{"post_id":24093452,"topic_id":2756545,"poster_id":783976,"post_rendered":"If you want to explain how you got your score you can comment! If you get higher than 20 comment your score!","post_canonical":"If you want to explain how you got your score you can comment! If you get higher than 20 comment your score!","username":"axusus","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":4,"nothanks_received":0,"thankers":"chessgocube, son7, megarnie, HWenslawski","deleted":false,"post_number":1,"post_time":1642044822,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_783976.jpeg?t=1638402869","num_posts":262,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2756545,"comment_count":238,"num_deleted":10,"topic_title":"What was your AMC 8 Score in 5th grade?","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":24093452,"first_poster_id":783976,"first_post_time":1642044822,"first_poster_name":"axusus","last_post_time":1653073913,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_783976.jpeg?t=1638402869","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_595281.png","last_post_id":25250635,"last_poster_id":595281,"last_poster_name":"heheman","last_update_time":1653073913,"category_id":3,"is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"tags":[{"tag_id":55,"tag_text":"AMC 8","is_visible":true},{"tag_id":29911,"tag_text":"poll","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"If you want to explain how you got your score you can comment! If you get higher than 20 comment your score!","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":1,"poll_id":44851,"source":"","category_num_users":13,"category_num_topics":46945,"category_num_posts":659613,"num_views":6472,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[4]={"2848786":{"num_posts":1,"posts_data":[{"post_id":25250572,"topic_id":2848786,"poster_id":167643,"post_rendered":"<b>VII.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.1 \/ S1.1<\/b> What is the parity of the number of ways to cut a rectangle into rectangles?<br>\n<br>\n<br>\n<b>1.2 \/ S1.2<\/b> In an isosceles triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > <span style=\"white-space:pre;\">(<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/9\/3c99bbe74cccdc92b249d376f13d9429d41a8b7a.png\" class=\"latex\" alt=\"$AB=AC$\" width=\"80\" height=\"13\" >)<\/span>. The circles <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > have different radii and lie outside the triangle. In this case, <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >.<\/span> Find the locus of the intersection of the common external tangents to these circles.<br>\n<br>\n<br>\n<b>1.5 \/ S1.5<\/b> There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)<br>\n<br>\n<br>\n<b>1.10 \/ S1.10<\/b> How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?<br>\n<br>\n<br>\n<b>2.1 \/ S2.1<\/b> Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.<br>\n<br>\n<br>\n<b>2.5 \/ S2.5<\/b> The plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > touches the sphere circumscribed around the tetrahedron <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >.<\/span> Prove that the lines of intersection of the planes of the faces <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/e\/6ce2cd123db24a7b8373999aea7a98927e3b3c81.png\" class=\"latex\" alt=\"$ACD$\" width=\"43\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/6\/a\/a6adfd3ad27a2dddeb0af7131d73c33c86991e93.png\" class=\"latex\" alt=\"$ABD$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > with the plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > divide it into six equal angles if and only if <img src=\"\/\/latex.artofproblemsolving.com\/0\/d\/6\/0d6daca6b8731859ee4ad0aaf1e473c1d920b8cb.png\" class=\"latex\" alt=\"$AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$\" style=\"vertical-align: -1px\" width=\"264\" height=\"14\" > .<br>\n<br>\n<br>\n<b>2.8 \/ S2.8<\/b> Is it possible to place a segment of length <img src=\"\/\/latex.artofproblemsolving.com\/4\/9\/4\/4944bcd1e5f1c6890c49160993314481c933217d.png\" class=\"latex\" alt=\"$1996$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" > on the checkered plane and choose a point <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > so that this segment does not touch nodes during any rotation around the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" >?<\/span><br>\n<br>\n<br>\n<b>2.9 \/ S2.9<\/b> The line intersects two concentric circles successively at the points <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/8\/a9874ea56cf8fec1bd246793224fbcfaad17ab3f.png\" class=\"latex\" alt=\"$A,B,C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > be the parallel chords of these circles, the points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > are the feet of the perpendiculars dropped from the points <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >,<\/span> respectively, on <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" >.<\/span> Prove that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/b\/5\/8b59ce9ac1bb33d8ddf80e77781727977923b0d0.png\" class=\"latex\" alt=\"$KF=ME$\" style=\"vertical-align: -1px\" width=\"91\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>3.3 \/ S3.2 <\/b>There are <img src=\"\/\/latex.artofproblemsolving.com\/0\/7\/0\/0705411d92671faf1fe5602cfef8f353c7f0f83d.png\" class=\"latex\" alt=\"$999$\" width=\"26\" height=\"12\" > points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?<br>\n<br>\n<br>\n<b>3.5<\/b> Let <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > be the center of a circle circumscribed about an acute triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >.<\/span> The centers of circles circumscribed around triangles <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/e\/a1e220396bf8f6704ea4a5f1745cc64ce339090b.png\" class=\"latex\" alt=\"$OAB$\" width=\"42\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/4\/3\/14323858397d132fc0169507e1fb72c5ae7ef54d.png\" class=\"latex\" alt=\"$OBC$\" width=\"43\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/a\/a\/3\/aa3c5323abf216d8a6208e043a0482849c8f045c.png\" class=\"latex\" alt=\"$OCA$\" width=\"42\" height=\"13\" > lie at the vertices of an equilateral triangle. Prove that the triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > is equilateral.<br>\n<br>\n<br>\n<b>3.7<\/b> In a trapezoid <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/b\/c\/5\/bc56862b1fdb71204894da278e42ef75de73a40b.png\" class=\"latex\" alt=\"$AD$\" width=\"28\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > are parallel with <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/9\/579ce9da0716efc125be9a95893c4c59d5f06713.png\" class=\"latex\" alt=\"$AD > BC$\" style=\"vertical-align: 0px\" width=\"82\" height=\"13\" >.<\/span> A point <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > is taken on the diagonal <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > such that <img src=\"\/\/latex.artofproblemsolving.com\/a\/7\/b\/a7baf95ecafc2d7791788ec79aeeae390d60b119.png\" class=\"latex\" alt=\"$BE$\" width=\"28\" height=\"12\" > is parallel to <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > . Prove that the areas of <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/7\/a\/b7a9191aefcf2eb237d1999486083ed78f6d515f.png\" class=\"latex\" alt=\"$DEC$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > are equal.<br>\n<br>\n<br>\n<b>3.10<\/b> A circle of radius <img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > is colored in two colors. Prove that there are two points of the same color at a distance of <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/2\/5\/5259ea0e230ae5bacea0d61a6861f37c3ec90016.png\" class=\"latex\" alt=\"$1.996$\" style=\"vertical-align: -1px\" width=\"43\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>4.3 \/ S4.6<\/b> Given a "parallel ruler", with which you can draw a straight line through <img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/c\/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.<br>\n<br>\n<br>\n<b>4.5<\/b> Let <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > be the intersection point of the angle bisectors <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/0\/9f0bd1bf44e9dd29cf5383c208c802b0dccb4920.png\" class=\"latex\" alt=\"$AA_1$\" style=\"vertical-align: -2px\" width=\"32\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/9\/9\/e99f9f70325371de6deea41a42eb2dccde9c8802.png\" class=\"latex\" alt=\"$BB_1$\" style=\"vertical-align: -2px\" width=\"34\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/1\/6\/f\/16f46c430dadb18e2cf3e9f17f6ed9e5e1138d48.png\" class=\"latex\" alt=\"$CC_1$\" style=\"vertical-align: -2px\" width=\"33\" height=\"15\" > of the triangle. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/f\/4\/d\/f4dc9e2e799b15fa08d14c2f85f6633c044c7ee6.png\" class=\"latexcenter\" alt=\"$$\\frac{AO}{OA_1} \\cdot \\frac{BO}{OB_1} \\cdot \\frac{CO}{OC_1} \\ge 8$$\" width=\"175\" height=\"40\" ><br>\n<br>\n<b>4.6 <\/b> Does there exist a closed <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/4\/b\/94bb6fb7a3ff722532798b45607025254679e7b2.png\" class=\"latex\" alt=\"$101$\" style=\"vertical-align: 0px\" width=\"26\" height=\"13\" >-<\/span>link polyline all of whose vertices have integer coordinates and all links of which have the same length?<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Senior Round have been posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c6h2848792p25250634\" class=\"bbcode_url\" target=\"_blank\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\" target=\"_blank\">here<\/a>.","post_canonical":"[b]VII.[\/b] [u]Rounds 1-4[\/u] \n\n\n[b]1.1 \/ S1.1[\/b] What is the parity of the number of ways to cut a rectangle into rectangles?\n\n\n[b]1.2 \/ S1.2[\/b] In an isosceles triangle $ABC$ ($AB=AC$). The circles $S_1$ and $S_2$ have different radii and lie outside the triangle. In this case, $S_1$ touches the line $AB$ at the point $A$, and $S_2$ touches the line $BC$ at the point $C$. Find the locus of the intersection of the common external tangents to these circles.\n\n\n[b]1.5 \/ S1.5[\/b] There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)\n\n\n[b]1.10 \/ S1.10[\/b] How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?\n\n\n[b]2.1 \/ S2.1[\/b] Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.\n\n\n[b]2.5 \/ S2.5[\/b] The plane $\\alpha$ touches the sphere circumscribed around the tetrahedron $ABCD$ at the point $A$. Prove that the lines of intersection of the planes of the faces $ABC$, $ACD$ and $ABD$ with the plane $\\alpha$ divide it into six equal angles if and only if $AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$ .\n\n\n[b]2.8 \/ S2.8[\/b] Is it possible to place a segment of length $1996$ on the checkered plane and choose a point $O$ so that this segment does not touch nodes during any rotation around the point $O$?\n\n\n[b]2.9 \/ S2.9[\/b] The line intersects two concentric circles successively at the points $A,B,C$ and $D$. Let $AE$ and $BF$ be the parallel chords of these circles, the points $K$ and $M$ are the feet of the perpendiculars dropped from the points $C$ and $D$, respectively, on $BF$ and $AE$. Prove that $KF=ME$.\n\n\n[b]3.3 \/ S3.2 [\/b]There are $999$ points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?\n\n\n[b]3.5[\/b] Let $O$ be the center of a circle circumscribed about an acute triangle $ABC$. The centers of circles circumscribed around triangles $OAB$, $OBC$, $OCA$ lie at the vertices of an equilateral triangle. Prove that the triangle $ABC$ is equilateral.\n\n\n[b]3.7[\/b] In a trapezoid $ABCD$, $AD$ and $BC$ are parallel with $AD > BC$. A point $E$ is taken on the diagonal $AC$ such that $BE$ is parallel to $CD$ . Prove that the areas of $ABC$ and $DEC$ are equal.\n\n\n[b]3.10[\/b] A circle of radius $1$ is colored in two colors. Prove that there are two points of the same color at a distance of $1.996$.\n\n\n[b]4.3 \/ S4.6[\/b] Given a \"parallel ruler\", with which you can draw a straight line through $2$ points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.\n\n\n[b]4.5[\/b] Let $O$ be the intersection point of the angle bisectors $AA_1$, $BB_1$, $CC_1$ of the triangle. Prove that $$\\frac{AO}{OA_1} \\cdot \\frac{BO}{OB_1} \\cdot \\frac{CO}{OC_1} \\ge 8$$\n\n\n[b]4.6 [\/b] Does there exist a closed $101$-link polyline all of whose vertices have integer coordinates and all links of which have the same length?\n\n\n\nPS. You should use hide for answers. Senior Round have been posted [url=https:\/\/artofproblemsolving.com\/community\/c6h2848792p25250634]here[\/url]. Collected [url=https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry]here[\/url].","username":"parmenides51","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1653073209,"num_edits":2,"post_format":"bbcode","last_edit_time":1653074075,"last_editor_username":"parmenides51","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","num_posts":18899,"editable":false,"deletable":true,"show_from_start":true,"show_from_end":true}],"topic_id":2848786,"comment_count":1,"num_deleted":0,"topic_title":"Geo from VII Russian Festival of Young Mathematicians 1996 Junior Rounds","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25250572,"first_poster_id":167643,"first_post_time":1653073209,"first_poster_name":"parmenides51","last_post_time":1653073209,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_post_id":25250572,"last_poster_id":167643,"last_poster_name":"parmenides51","last_update_time":1653074075,"category_id":4,"is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":146,"tag_text":"3D geometry","is_visible":true},{"tag_id":1600924,"tag_text":"RFYM","is_visible":true},{"tag_id":41772,"tag_text":"combinatorial geometry","is_visible":true},{"tag_id":267,"tag_text":"geometric inequality","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"<b>VII.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.1 \/ S1.1<\/b> What is the parity of the number of ways to cut a rectangle into rectangles?<br>\n<br>\n<br>\n<b>1.2 \/ S1.2<\/b> In an isosceles triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > <span style=\"white-space:pre;\">(<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/9\/3c99bbe74cccdc92b249d376f13d9429d41a8b7a.png\" class=\"latex\" alt=\"$AB=AC$\" width=\"80\" height=\"13\" >)<\/span>. The circles <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > have different radii and lie outside the triangle. In this case, <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >.<\/span> Find the locus of the intersection of the common external tangents to these circles.<br>\n<br>\n<br>\n<b>1.5 \/ S1.5<\/b> There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)<br>\n<br>\n<br>\n<b>1.10 \/ S1.10<\/b> How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?<br>\n<br>\n<br>\n<b>2.1 \/ S2.1<\/b> Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.<br>\n<br>\n<br>\n<b>2.5 \/ S2.5<\/b> The plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > touches the sphere circumscribed around the tetrahedron <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >.<\/span> Prove that the lines of intersection of the planes of the faces <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/e\/6ce2cd123db24a7b8373999aea7a98927e3b3c81.png\" class=\"latex\" alt=\"$ACD$\" width=\"43\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/6\/a\/a6adfd3ad27a2dddeb0af7131d73c33c86991e93.png\" class=\"latex\" alt=\"$ABD$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > with the plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > divide it into six equal angles if and only if <img src=\"\/\/latex.artofproblemsolving.com\/0\/d\/6\/0d6daca6b8731859ee4ad0aaf1e473c1d920b8cb.png\" class=\"latex\" alt=\"$AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$\" style=\"vertical-align: -1px\" width=\"264\" height=\"14\" > .<br>\n<br>\n<br>\n<b>2.8 \/ S2.8<\/b> Is it possible to place a segment of length <img src=\"\/\/latex.artofproblemsolving.com\/4\/9\/4\/4944bcd1e5f1c6890c49160993314481c933217d.png\" class=\"latex\" alt=\"$1996$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" > on the checkered plane and choose a point <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > so that this segment does not touch nodes during any rotation around the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" >?<\/span><br>\n<br>\n<br>\n<b>2.9 \/ S2.9<\/b> The line intersects two concentric circles successively at the points <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/8\/a9874ea56cf8fec1bd246793224fbcfaad17ab3f.png\" class=\"latex\" alt=\"$A,B,C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > be the parallel chords of these circles, the points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > are the feet of the perpendiculars dropped from the points <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >,<\/span> respectively, on <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" >.<\/span> Prove that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/b\/5\/8b59ce9ac1bb33d8ddf80e77781727977923b0d0.png\" class=\"latex\" alt=\"$KF=ME$\" style=\"vertical-align: -1px\" width=\"91\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>3.3 \/ S3.2 <\/b>There are <img src=\"\/\/latex.artofproblemsolving.com\/0\/7\/0\/0705411d92671faf1fe5602cfef8f353c7f0f83d.png\" class=\"latex\" alt=\"$999$\" width=\"26\" height=\"12\" > points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?<br>\n<br>\n<br>\n<b>3.5<\/b> Let <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > be the center of a circle circumscribed about an acute triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >.<\/span> The centers of circles circumscribed around triangles <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/e\/a1e220396bf8f6704ea4a5f1745cc64ce339090b.png\" class=\"latex\" alt=\"$OAB$\" width=\"42\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/4\/3\/14323858397d132fc0169507e1fb72c5ae7ef54d.png\" class=\"latex\" alt=\"$OBC$\" width=\"43\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/a\/a\/3\/aa3c5323abf216d8a6208e043a0482849c8f045c.png\" class=\"latex\" alt=\"$OCA$\" width=\"42\" height=\"13\" > lie at the vertices of an equilateral triangle. Prove that the triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > is equilateral.<br>\n<br>\n<br>\n<b>3.7<\/b> In a trapezoid <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/b\/c\/5\/bc56862b1fdb71204894da278e42ef75de73a40b.png\" class=\"latex\" alt=\"$AD$\" width=\"28\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > are parallel with <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/9\/579ce9da0716efc125be9a95893c4c59d5f06713.png\" class=\"latex\" alt=\"$AD > BC$\" style=\"vertical-align: 0px\" width=\"82\" height=\"13\" >.<\/span> A point <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > is taken on the diagonal <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > such that <img src=\"\/\/latex.artofproblemsolving.com\/a\/7\/b\/a7baf95ecafc2d7791788ec79aeeae390d60b119.png\" class=\"latex\" alt=\"$BE$\" width=\"28\" height=\"12\" > is parallel to <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > . Prove that the areas of <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/7\/a\/b7a9191aefcf2eb237d1999486083ed78f6d515f.png\" class=\"latex\" alt=\"$DEC$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > are equal.<br>\n<br>\n<br>\n<b>3.10<\/b> A circle of radius <img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > is colored in two colors. Prove that there are two points of the same color at a distance of <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/2\/5\/5259ea0e230ae5bacea0d61a6861f37c3ec90016.png\" class=\"latex\" alt=\"$1.996$\" style=\"vertical-align: -1px\" width=\"43\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>4.3 \/ S4.6<\/b> Given a "parallel ruler", with which you can draw a straight line through <img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/c\/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.<br>\n<br>\n<br>\n<b>4.5<\/b> Let <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > be the intersection point of the angle bisectors <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/0\/9f0bd1bf44e9dd29cf5383c208c802b0dccb4920.png\" class=\"latex\" alt=\"$AA_1$\" style=\"vertical-align: -2px\" width=\"32\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/9\/9\/e99f9f70325371de6deea41a42eb2dccde9c8802.png\" class=\"latex\" alt=\"$BB_1$\" style=\"vertical-align: -2px\" width=\"34\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/1\/6\/f\/16f46c430dadb18e2cf3e9f17f6ed9e5e1138d48.png\" class=\"latex\" alt=\"$CC_1$\" style=\"vertical-align: -2px\" width=\"33\" height=\"15\" > of the triangle. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/f\/4\/d\/f4dc9e2e799b15fa08d14c2f85f6633c044c7ee6.png\" class=\"latexcenter\" alt=\"$$\\frac{AO}{OA_1} \\cdot \\frac{BO}{OB_1} \\cdot \\frac{CO}{OC_1} \\ge 8$$\" width=\"175\" height=\"40\" ><br>\n<br>\n<b>4.6 <\/b> Does there exist a closed <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/4\/b\/94bb6fb7a3ff722532798b45607025254679e7b2.png\" class=\"latex\" alt=\"$101$\" style=\"vertical-align: 0px\" width=\"26\" height=\"13\" >-<\/span>link polyline all of whose vertices have integer coordinates and all links of which have the same length?<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Senior Round have been posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c6h2848792p25250634\" class=\"bbcode_url\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\">here<\/a>.","category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","num_reports":0,"poll_id":0,"source":"","category_num_users":6,"category_num_topics":103886,"category_num_posts":565536,"num_views":7,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":true,"in_feed":true,"is_watched":true},"2844078":{"num_posts":4,"posts_data":[{"post_id":25198764,"topic_id":2844078,"poster_id":244795,"post_rendered":"How many the number of arrangements of the letters AAAABBBC in which either the A's appear together in a block of four letters or the B's appear together in a block of three letters?<br>\n<br>\nCould you tell the answer and the explanation?","post_canonical":"How many the number of arrangements of the letters AAAABBBC in which either the A's appear together in a block of four letters or the B's appear together in a block of three letters?\n\nCould you tell the answer and the explanation?","username":"bsming","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1652436811,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_244795.png","num_posts":35,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2844078,"comment_count":4,"num_deleted":0,"topic_title":"arrangements of the letters AAAABBBC","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25198764,"first_poster_id":244795,"first_post_time":1652436811,"first_poster_name":"bsming","last_post_time":1653069633,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_244795.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1313.png","last_post_id":25250283,"last_poster_id":1313,"last_poster_name":"gauss202","last_update_time":1653069633,"category_id":4,"is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"tags":[{"tag_id":139072,"tag_text":"MATHCOUNTS prep","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"How many the number of arrangements of the letters AAAABBBC in which either the A's appear together in a block of four letters or the B's appear together in a block of three letters?<br>\n<br>\nCould you tell the answer and the explanation?","category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","num_reports":0,"poll_id":0,"source":"","category_num_users":6,"category_num_topics":103886,"category_num_posts":565536,"num_views":253,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848183":{"num_posts":2,"posts_data":[{"post_id":25243640,"topic_id":2848183,"poster_id":167643,"post_rendered":"In triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > with angles <img src=\"\/\/latex.artofproblemsolving.com\/6\/0\/6\/606e48e5ea682394ddf3cae870f5fc5bdcd98299.png\" class=\"latex\" alt=\"$\\angle ACB = 40^o$\" style=\"vertical-align: -1px\" width=\"107\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/d\/d\/bddbb513df793a91f335bfb77ba43d14bcd55a4b.png\" class=\"latex\" alt=\"$\\angle BAC = 60^o$\" width=\"104\" height=\"13\" > on segment <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> a point <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is chosen such that <img src=\"\/\/latex.artofproblemsolving.com\/0\/e\/0\/0e0643fdf9f0062421fc69967aa84bfbfd63bab1.png\" class=\"latex\" alt=\"$2 \\cdot CD = AB$\" style=\"vertical-align: -1px\" width=\"106\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is the midpoint of segment <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >.<\/span> Find <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/e\/7\/de73b9742eac7d4adfff2b13361e3bceaac9c8f9.png\" class=\"latex\" alt=\"$\\angle CMD$\" width=\"61\" height=\"12\" >.<\/span>","post_canonical":"In triangle $ABC$ with angles $\\angle ACB = 40^o$ and $\\angle BAC = 60^o$ on segment $BC$, a point $D$ is chosen such that $2 \\cdot CD = AB$ and $M$ is the midpoint of segment $AC$. Find $\\angle CMD$.","username":"parmenides51","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1652986328,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","num_posts":18899,"editable":false,"deletable":true,"show_from_start":true,"show_from_end":false}],"topic_id":2848183,"comment_count":2,"num_deleted":0,"topic_title":"<CMD=? 100-60-40 triangle, 2CD=AB 2022 Russian Young Mathematician 6-7.8","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25243640,"first_poster_id":167643,"first_post_time":1652986328,"first_poster_name":"parmenides51","last_post_time":1653069099,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_36948.jpg","last_post_id":25250239,"last_poster_id":36948,"last_poster_name":"sunken rock","last_update_time":1653069099,"category_id":4,"is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":45165,"tag_text":"angles","is_visible":true},{"tag_id":1629561,"tag_text":"Young Mathematician","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"In triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > with angles <img src=\"\/\/latex.artofproblemsolving.com\/6\/0\/6\/606e48e5ea682394ddf3cae870f5fc5bdcd98299.png\" class=\"latex\" alt=\"$\\angle ACB = 40^o$\" style=\"vertical-align: -1px\" width=\"107\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/d\/d\/bddbb513df793a91f335bfb77ba43d14bcd55a4b.png\" class=\"latex\" alt=\"$\\angle BAC = 60^o$\" width=\"104\" height=\"13\" > on segment <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> a point <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is chosen such that <img src=\"\/\/latex.artofproblemsolving.com\/0\/e\/0\/0e0643fdf9f0062421fc69967aa84bfbfd63bab1.png\" class=\"latex\" alt=\"$2 \\cdot CD = AB$\" style=\"vertical-align: -1px\" width=\"106\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is the midpoint of segment <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >.<\/span> Find <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/e\/7\/de73b9742eac7d4adfff2b13361e3bceaac9c8f9.png\" class=\"latex\" alt=\"$\\angle CMD$\" width=\"61\" height=\"12\" >.<\/span>","category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","num_reports":0,"poll_id":0,"source":"","category_num_users":6,"category_num_topics":103886,"category_num_posts":565536,"num_views":76,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":true,"in_feed":true,"is_watched":true},"2835578":{"num_posts":2,"posts_data":[{"post_id":25102300,"topic_id":2835578,"poster_id":167643,"post_rendered":"<b>VI.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.2<\/b> <img src=\"\/\/latex.artofproblemsolving.com\/b\/f\/2\/bf2c9074b396e3af0dea52d792660eea1c77f10f.png\" class=\"latex\" alt=\"$9$\" width=\"8\" height=\"12\" > numbers are written out, the lengths of the angle bisectors, altitudes and medians of a certain triangle. It is known that among them there are no more than <img src=\"\/\/latex.artofproblemsolving.com\/c\/7\/c\/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png\" class=\"latex\" alt=\"$4$\" style=\"vertical-align: 0px\" width=\"9\" height=\"12\" > different ones. Prove that this triangle is isosceles.<br>\n<br>\n<br>\n<b>1.3<\/b> Is it possible to cut a cube into 6 equal triangular pyramids?<br>\n<br>\n<br>\n<b>1.6 <\/b> All points of the strip of width 0.001 are colored in two colors. Prove that there are two points of the same color at a distance of 1.<br>\n<br>\n<br>\n<b>1.7<\/b> On sides <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/e\/6\/5\/e657259c0995d2c36d46744bb4aa81471e6575ec.png\" class=\"latex\" alt=\"$CA$\" width=\"28\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > of triangle ABC, points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/c\/36c7f3f828cf12ab9cafeffad282468a95219600.png\" class=\"latex\" alt=\"$A_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/2\/4\/3\/243af6fad28c15abb3e11996374ba60e745bf543.png\" class=\"latex\" alt=\"$B_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/b\/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png\" class=\"latex\" alt=\"$C_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > are taken respectively so that <img src=\"\/\/latex.artofproblemsolving.com\/3\/d\/9\/3d986133b5881edc28896966ce4e393c1c4907f1.png\" class=\"latexcenter\" alt=\"$$AC_1:C_1B = BA_1:A_1C = CB_1:B_1A = 2:1$$\" width=\"360\" height=\"16\" >Prove that if triangle <img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/1\/9b1815ca31a0a4b18b725b690862a84de67aed84.png\" class=\"latex\" alt=\"$A_1B_1C_1$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" > is equilateral, then triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > is also equilateral.<br>\n<br>\n<br>\n<b>2.5<\/b> On sides <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/3\/0b31cd391bebc36897a8c10ea78e5d584980b049.png\" class=\"latex\" alt=\"$DA$\" width=\"29\" height=\"13\" > of parallelogram <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> points <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/0\/5\/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" > are taken, respectively. Let <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > be the intersection point of segments <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/7\/d\/07d29465728e92952374a857beaee91665d8dbd9.png\" class=\"latex\" alt=\"$CF$\" style=\"vertical-align: -1px\" width=\"30\" height=\"14\" >.<\/span> Prove that if the areas of triangles <img src=\"\/\/latex.artofproblemsolving.com\/c\/e\/6\/ce6bc740cb6cb591e4787222c09218c7130355f1.png\" class=\"latex\" alt=\"$AKF$\" width=\"44\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/c\/d\/b\/cdbde756c78aa0f706791826b72f87f17a154349.png\" class=\"latex\" alt=\"$CKE$\" style=\"vertical-align: -1px\" width=\"47\" height=\"14\" > are equal, then the point <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > lies on the diagonal <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" >.<\/span><br>\n<br>\n<br>\n<b>2.8<\/b> Find the smallest possible length of a simple closed polyline that has at least one common point with each face of the unit cube.<br>\n<br>\n<br>\n<b>3.1<\/b> Six circles have a common interior point. Prove that the center of one of them lies inside the other.<br>\n<br>\n<br>\n<b>3.3<\/b> An arbitrary point <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is taken inside the regular hexagon <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/0\/2\/8029902caf855d940e465d04ff40101d8850c3e0.png\" class=\"latex\" alt=\"$ABCDEF$\" width=\"86\" height=\"13\" >.<\/span> Prove that it is possible to form a hexagon from the segments <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/2\/f\/62f7eb82948ac9d7e8659abb37508b1015d8a3cf.png\" class=\"latex\" alt=\"$MA$\" width=\"33\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/6\/856c130e0bb237c3edeb7e3c0246d40cc607e861.png\" class=\"latex\" alt=\"$MB$\" width=\"33\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/d\/5\/4d5ee546f7ea84755eae7a58fbf1d892bd258118.png\" class=\"latex\" alt=\"$MC$\" width=\"33\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/2\/f\/f2f00f55b6b1cf59b8a3b3687a2b9aed8b37578c.png\" class=\"latex\" alt=\"$MD$\" width=\"34\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/1\/a\/d1a30c9c946a561bfb130894432aa55948d58b0d.png\" class=\"latex\" alt=\"$ME$\" style=\"vertical-align: -1px\" width=\"36\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/5\/1\/551fab3023436463570c8159f37571740d523aa7.png\" class=\"latex\" alt=\"$MF$\" style=\"vertical-align: -1px\" width=\"36\" height=\"14\" >,<\/span> the area of which is not less than <img src=\"\/\/latex.artofproblemsolving.com\/6\/1\/1\/611d08a8c57a214b2b964bba16da8157a5b81c9a.png\" class=\"latex\" alt=\"$\\frac23$\" style=\"vertical-align: -13px\" width=\"15\" height=\"38\" > of the area of the original hexagon.<br>\n<br>\n<br>\n<b>3.5<\/b> Given a triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> all angles of which are expressed in integer degrees and are different from <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/6\/2\/b6209f33ef77435a57e3b1ab0bbe2185e7f26863.png\" class=\"latex\" alt=\"$45^o$\" style=\"vertical-align: 0px\" width=\"24\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/9\/37943786559be7f13b0927b59e138ce1ace0eea6.png\" class=\"latex\" alt=\"$90^o$\" style=\"vertical-align: -1px\" width=\"27\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/e\/ade07d1dcbcdf5109752313df61e920d192ca8c0.png\" class=\"latex\" alt=\"$135^o$\" style=\"vertical-align: 0px\" width=\"33\" height=\"13\" >.<\/span> Points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/c\/36c7f3f828cf12ab9cafeffad282468a95219600.png\" class=\"latex\" alt=\"$A_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/2\/4\/3\/243af6fad28c15abb3e11996374ba60e745bf543.png\" class=\"latex\" alt=\"$B_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/b\/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png\" class=\"latex\" alt=\"$C_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > are the feet of its altitudes, points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/3\/a\/e3a8ae66ded7f33f9385a37c9cadfb82720013d8.png\" class=\"latex\" alt=\"$A_2$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/8\/368b6f971b418d731c62ff2cb59f51c527d22849.png\" class=\"latex\" alt=\"$B_2$\" style=\"vertical-align: -2px\" width=\"20\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/b\/6\/5\/b65e6facfe122c182fe14d503aeace2c52a9739d.png\" class=\"latex\" alt=\"$C_2$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > are the feet of the altitudes of triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/1\/9b1815ca31a0a4b18b725b690862a84de67aed84.png\" class=\"latex\" alt=\"$A_1B_1C_1$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" >,<\/span> etc. . Prove that there are infinitely many triangles similar to each other.<br>\n<br>\n<br>\n<b>4.2<\/b> The circle with center <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > is tangent to the sides of the angle with vertex <img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" >.<\/span> The tangent to the circle intersects line segments <img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/0\/370334b1a3582d54fad42c735dfdded7d216d3cc.png\" class=\"latex\" alt=\"$AK$\" width=\"30\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/e\/7\/3\/e73bc78b745edf7fe1e55dbfc3d914190faf5015.png\" class=\"latex\" alt=\"$AM$\" width=\"32\" height=\"13\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/f\/f\/5\/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" style=\"vertical-align: -1px\" width=\"17\" height=\"14\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >,<\/span> respectively, and the line KM intersects line segments <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/6\/0964c9cf406ef8e5ec8beaa6a92125f2d8f90460.png\" class=\"latex\" alt=\"$OB$\" width=\"28\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/2\/0\/f\/20f8f2f1a3ddc6b42fae25a0637e85c747875180.png\" class=\"latex\" alt=\"$OC$\" width=\"28\" height=\"12\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" >.<\/span> Prove that the area of triangle <img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/f\/adfbfed38bc145be608628b64295119ef073af15.png\" class=\"latex\" alt=\"$ODE$\" width=\"44\" height=\"12\" > is equal to a quarter of the area of triangle <img src=\"\/\/latex.artofproblemsolving.com\/7\/6\/d\/76d9b534de592b2a7fc63b97e1f0faab799204e9.png\" class=\"latex\" alt=\"$BOC$\" width=\"43\" height=\"12\" > if and only if angle A is <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/e\/4ae82fc01ad7a31ae4c8a590a57a1fc4f150d228.png\" class=\"latex\" alt=\"$60^o$\" width=\"24\" height=\"12\" >.<\/span><br>\n<br>\n<br>\n<b>4.3<\/b> In the middle of a round lake there is an island (a convex figure), which is visible from any point on the shore of the lake at an angle of <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/e\/4ae82fc01ad7a31ae4c8a590a57a1fc4f150d228.png\" class=\"latex\" alt=\"$60^o$\" width=\"24\" height=\"12\" >.<\/span> Prove that the island has the shape of a circle.<br>\n<br>\n<br>\n<b>4.5<\/b> Several (finite) sides of the cells of the endless checkered paper are colored red. Once a second, all nodes are selected, each of which has at least two red segments coming out, and all other segments coming out of these nodes are also painted red. Prove that the number of red segments cannot increase indefinitely.<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Senior Round posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c6h2835588p25102426\" class=\"bbcode_url\" target=\"_blank\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\" target=\"_blank\">here<\/a>.","post_canonical":"[b]VI.[\/b] [u]Rounds 1-4[\/u] \n\n\n[b]1.2[\/b] $9$ numbers are written out, the lengths of the angle bisectors, altitudes and medians of a certain triangle. It is known that among them there are no more than $4$ different ones. Prove that this triangle is isosceles.\n\n\n[b]1.3[\/b] Is it possible to cut a cube into 6 equal triangular pyramids?\n\n\n[b]1.6 [\/b] All points of the strip of width 0.001 are colored in two colors. Prove that there are two points of the same color at a distance of 1.\n\n\n[b]1.7[\/b] On sides $BC$, $CA$ and $AB$ of triangle ABC, points $A_1$, $B_1$ and $C_1$ are taken respectively so that $$AC_1:C_1B = BA_1:A_1C = CB_1:B_1A = 2:1$$ Prove that if triangle $A_1B_1C_1$ is equilateral, then triangle $ABC$ is also equilateral.\n\n\n[b]2.5[\/b] On sides $CD$ and $DA$ of parallelogram $ABCD$, points $E$ and $F$ are taken, respectively. Let $K$ be the intersection point of segments $AE$ and $CF$. Prove that if the areas of triangles $AKF$ and $CKE$ are equal, then the point $K$ lies on the diagonal $BD$.\n\n\n[b]2.8[\/b] Find the smallest possible length of a simple closed polyline that has at least one common point with each face of the unit cube.\n\n\n[b]3.1[\/b] Six circles have a common interior point. Prove that the center of one of them lies inside the other.\n\n\n[b]3.3[\/b] An arbitrary point $M$ is taken inside the regular hexagon $ABCDEF$. Prove that it is possible to form a hexagon from the segments $MA$, $MB$, $MC$, $MD$, $ME$, $MF$, the area of \u200b\u200bwhich is not less than $\\frac23$ of the area of \u200b\u200bthe original hexagon.\n\n\n[b]3.5[\/b] Given a triangle $ABC$, all angles of which are expressed in integer degrees and are different from $45^o$, $90^o$, $135^o$. Points $A_1$, $B_1$, $C_1$ are the feet of its altitudes, points $A_2$, $B_2$, $C_2$ are the feet of the altitudes of triangle $A_1B_1C_1$, etc. . Prove that there are infinitely many triangles similar to each other.\n\n\n[b]4.2[\/b] The circle with center $O$ is tangent to the sides of the angle with vertex $A$ at points $K$ and $M$. The tangent to the circle intersects line segments $AK$ and $AM$ at points $B$ and $C$, respectively, and the line KM intersects line segments $OB$ and $OC$ at points $D$ and $E$. Prove that the area of \u200b\u200btriangle $ODE$ is equal to a quarter of the area of \u200b\u200btriangle $BOC$ if and only if angle A is $60^o$.\n\n\n[b]4.3[\/b] In the middle of a round lake there is an island (a convex figure), which is visible from any point on the shore of the lake at an angle of $60^o$. Prove that the island has the shape of a circle.\n\n\n[b]4.5[\/b] Several (finite) sides of the cells of the endless checkered paper are colored red. Once a second, all nodes are selected, each of which has at least two red segments coming out, and all other segments coming out of these nodes are also painted red. Prove that the number of red segments cannot increase indefinitely.\n\n\n\nPS. You should use hide for answers. Senior Round posted [url=https:\/\/artofproblemsolving.com\/community\/c6h2835588p25102426]here[\/url]. Collected [url=https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry]here[\/url].","username":"parmenides51","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1651355815,"num_edits":8,"post_format":"bbcode","last_edit_time":1653068704,"last_editor_username":"parmenides51","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","num_posts":18899,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2835578,"comment_count":2,"num_deleted":0,"topic_title":"Geo from VI Russian Festival of Young Mathematicians 1995 Junior Rounds","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25102300,"first_poster_id":167643,"first_post_time":1651355815,"first_poster_name":"parmenides51","last_post_time":1653068725,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_post_id":25250206,"last_poster_id":167643,"last_poster_name":"parmenides51","last_update_time":1653068725,"category_id":4,"is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":1600924,"tag_text":"RFYM","is_visible":true},{"tag_id":146,"tag_text":"3D geometry","is_visible":true},{"tag_id":41772,"tag_text":"combinatorial geometry","is_visible":true},{"tag_id":267,"tag_text":"geometric inequality","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"<b>VI.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.2<\/b> <img src=\"\/\/latex.artofproblemsolving.com\/b\/f\/2\/bf2c9074b396e3af0dea52d792660eea1c77f10f.png\" class=\"latex\" alt=\"$9$\" width=\"8\" height=\"12\" > numbers are written out, the lengths of the angle bisectors, altitudes and medians of a certain triangle. It is known that among them there are no more than <img src=\"\/\/latex.artofproblemsolving.com\/c\/7\/c\/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png\" class=\"latex\" alt=\"$4$\" style=\"vertical-align: 0px\" width=\"9\" height=\"12\" > different ones. Prove that this triangle is isosceles.<br>\n<br>\n<br>\n<b>1.3<\/b> Is it possible to cut a cube into 6 equal triangular pyramids?<br>\n<br>\n<br>\n<b>1.6 <\/b> All points of the strip of width 0.001 are colored in two colors. Prove that there are two points of the same color at a distance of 1.<br>\n<br>\n<br>\n<b>1.7<\/b> On sides <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/e\/6\/5\/e657259c0995d2c36d46744bb4aa81471e6575ec.png\" class=\"latex\" alt=\"$CA$\" width=\"28\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > of triangle ABC, points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/c\/36c7f3f828cf12ab9cafeffad282468a95219600.png\" class=\"latex\" alt=\"$A_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/2\/4\/3\/243af6fad28c15abb3e11996374ba60e745bf543.png\" class=\"latex\" alt=\"$B_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/b\/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png\" class=\"latex\" alt=\"$C_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > are taken respectively so that <img src=\"\/\/latex.artofproblemsolving.com\/3\/d\/9\/3d986133b5881edc28896966ce4e393c1c4907f1.png\" class=\"latexcenter\" alt=\"$$AC_1:C_1B = BA_1:A_1C = CB_1:B_1A = 2:1$$\" width=\"360\" height=\"16\" >Prove that if triangle <img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/1\/9b1815ca31a0a4b18b725b690862a84de67aed84.png\" class=\"latex\" alt=\"$A_1B_1C_1$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" > is equilateral, then triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > is also equilateral.<br>\n<br>\n<br>\n<b>2.5<\/b> On sides <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/3\/0b31cd391bebc36897a8c10ea78e5d584980b049.png\" class=\"latex\" alt=\"$DA$\" width=\"29\" height=\"13\" > of parallelogram <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> points <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/0\/5\/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" > are taken, respectively. Let <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > be the intersection point of segments <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/7\/d\/07d29465728e92952374a857beaee91665d8dbd9.png\" class=\"latex\" alt=\"$CF$\" style=\"vertical-align: -1px\" width=\"30\" height=\"14\" >.<\/span> Prove that if the areas of triangles <img src=\"\/\/latex.artofproblemsolving.com\/c\/e\/6\/ce6bc740cb6cb591e4787222c09218c7130355f1.png\" class=\"latex\" alt=\"$AKF$\" width=\"44\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/c\/d\/b\/cdbde756c78aa0f706791826b72f87f17a154349.png\" class=\"latex\" alt=\"$CKE$\" style=\"vertical-align: -1px\" width=\"47\" height=\"14\" > are equal, then the point <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > lies on the diagonal <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" >.<\/span><br>\n<br>\n<br>\n<b>2.8<\/b> Find the smallest possible length of a simple closed polyline that has at least one common point with each face of the unit cube.<br>\n<br>\n<br>\n<b>3.1<\/b> Six circles have a common interior point. Prove that the center of one of them lies inside the other.<br>\n<br>\n<br>\n<b>3.3<\/b> An arbitrary point <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is taken inside the regular hexagon <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/0\/2\/8029902caf855d940e465d04ff40101d8850c3e0.png\" class=\"latex\" alt=\"$ABCDEF$\" width=\"86\" height=\"13\" >.<\/span> Prove that it is possible to form a hexagon from the segments <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/2\/f\/62f7eb82948ac9d7e8659abb37508b1015d8a3cf.png\" class=\"latex\" alt=\"$MA$\" width=\"33\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/6\/856c130e0bb237c3edeb7e3c0246d40cc607e861.png\" class=\"latex\" alt=\"$MB$\" width=\"33\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/d\/5\/4d5ee546f7ea84755eae7a58fbf1d892bd258118.png\" class=\"latex\" alt=\"$MC$\" width=\"33\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/2\/f\/f2f00f55b6b1cf59b8a3b3687a2b9aed8b37578c.png\" class=\"latex\" alt=\"$MD$\" width=\"34\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/1\/a\/d1a30c9c946a561bfb130894432aa55948d58b0d.png\" class=\"latex\" alt=\"$ME$\" style=\"vertical-align: -1px\" width=\"36\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/5\/1\/551fab3023436463570c8159f37571740d523aa7.png\" class=\"latex\" alt=\"$MF$\" style=\"vertical-align: -1px\" width=\"36\" height=\"14\" >,<\/span> the area of which is not less than <img src=\"\/\/latex.artofproblemsolving.com\/6\/1\/1\/611d08a8c57a214b2b964bba16da8157a5b81c9a.png\" class=\"latex\" alt=\"$\\frac23$\" style=\"vertical-align: -13px\" width=\"15\" height=\"38\" > of the area of the original hexagon.<br>\n<br>\n<br>\n<b>3.5<\/b> Given a triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> all angles of which are expressed in integer degrees and are different from <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/6\/2\/b6209f33ef77435a57e3b1ab0bbe2185e7f26863.png\" class=\"latex\" alt=\"$45^o$\" style=\"vertical-align: 0px\" width=\"24\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/9\/37943786559be7f13b0927b59e138ce1ace0eea6.png\" class=\"latex\" alt=\"$90^o$\" style=\"vertical-align: -1px\" width=\"27\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/e\/ade07d1dcbcdf5109752313df61e920d192ca8c0.png\" class=\"latex\" alt=\"$135^o$\" style=\"vertical-align: 0px\" width=\"33\" height=\"13\" >.<\/span> Points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/c\/36c7f3f828cf12ab9cafeffad282468a95219600.png\" class=\"latex\" alt=\"$A_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/2\/4\/3\/243af6fad28c15abb3e11996374ba60e745bf543.png\" class=\"latex\" alt=\"$B_1$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/b\/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png\" class=\"latex\" alt=\"$C_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > are the feet of its altitudes, points <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/3\/a\/e3a8ae66ded7f33f9385a37c9cadfb82720013d8.png\" class=\"latex\" alt=\"$A_2$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/8\/368b6f971b418d731c62ff2cb59f51c527d22849.png\" class=\"latex\" alt=\"$B_2$\" style=\"vertical-align: -2px\" width=\"20\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/b\/6\/5\/b65e6facfe122c182fe14d503aeace2c52a9739d.png\" class=\"latex\" alt=\"$C_2$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > are the feet of the altitudes of triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/1\/9b1815ca31a0a4b18b725b690862a84de67aed84.png\" class=\"latex\" alt=\"$A_1B_1C_1$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" >,<\/span> etc. . Prove that there are infinitely many triangles similar to each other.<br>\n<br>\n<br>\n<b>4.2<\/b> The circle with center <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > is tangent to the sides of the angle with vertex <img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" >.<\/span> The tangent to the circle intersects line segments <img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/0\/370334b1a3582d54fad42c735dfdded7d216d3cc.png\" class=\"latex\" alt=\"$AK$\" width=\"30\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/e\/7\/3\/e73bc78b745edf7fe1e55dbfc3d914190faf5015.png\" class=\"latex\" alt=\"$AM$\" width=\"32\" height=\"13\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/f\/f\/5\/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" style=\"vertical-align: -1px\" width=\"17\" height=\"14\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >,<\/span> respectively, and the line KM intersects line segments <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/6\/0964c9cf406ef8e5ec8beaa6a92125f2d8f90460.png\" class=\"latex\" alt=\"$OB$\" width=\"28\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/2\/0\/f\/20f8f2f1a3ddc6b42fae25a0637e85c747875180.png\" class=\"latex\" alt=\"$OC$\" width=\"28\" height=\"12\" > at points <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" >.<\/span> Prove that the area of triangle <img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/f\/adfbfed38bc145be608628b64295119ef073af15.png\" class=\"latex\" alt=\"$ODE$\" width=\"44\" height=\"12\" > is equal to a quarter of the area of triangle <img src=\"\/\/latex.artofproblemsolving.com\/7\/6\/d\/76d9b534de592b2a7fc63b97e1f0faab799204e9.png\" class=\"latex\" alt=\"$BOC$\" width=\"43\" height=\"12\" > if and only if angle A is <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/e\/4ae82fc01ad7a31ae4c8a590a57a1fc4f150d228.png\" class=\"latex\" alt=\"$60^o$\" width=\"24\" height=\"12\" >.<\/span><br>\n<br>\n<br>\n<b>4.3<\/b> In the middle of a round lake there is an island (a convex figure), which is visible from any point on the shore of the lake at an angle of <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/e\/4ae82fc01ad7a31ae4c8a590a57a1fc4f150d228.png\" class=\"latex\" alt=\"$60^o$\" width=\"24\" height=\"12\" >.<\/span> Prove that the island has the shape of a circle.<br>\n<br>\n<br>\n<b>4.5<\/b> Several (finite) sides of the cells of the endless checkered paper are colored red. Once a second, all nodes are selected, each of which has at least two red segments coming out, and all other segments coming out of these nodes are also painted red. Prove that the number of red segments cannot increase indefinitely.<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Senior Round posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c6h2835588p25102426\" class=\"bbcode_url\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\">here<\/a>.","category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","num_reports":0,"poll_id":0,"source":"","category_num_users":6,"category_num_topics":103886,"category_num_posts":565536,"num_views":117,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":true,"in_feed":true,"is_watched":true}};AoPS.preload_topics[5]={"2839375":{"num_posts":81,"posts_data":[{"post_id":25146194,"topic_id":2839375,"poster_id":399854,"post_rendered":"Does anyone know when the decisions for HCSSiM will come out this year?<br>\n<br>\nPlease post info if you received the decision letter.","post_canonical":"Does anyone know when the decisions for HCSSiM will come out this year?\n\nPlease post info if you received the decision letter.\n\n","username":"jumpee73","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":3,"nothanks_received":0,"thankers":"HWenslawski, ImSh95, MasterInTheMaking","deleted":false,"post_number":1,"post_time":1651797539,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_399854.png?t=1652466871","num_posts":3,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2839375,"comment_count":81,"num_deleted":0,"topic_title":"HCSSiM 2022 decisions?","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25146194,"first_poster_id":399854,"first_post_time":1651797539,"first_poster_name":"jumpee73","last_post_time":1653073707,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_399854.png?t=1652466871","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_595281.png","last_post_id":25250614,"last_poster_id":595281,"last_poster_name":"heheman","last_update_time":1653073707,"category_id":5,"is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"tags":[{"tag_id":62,"tag_text":"HCSSiM","is_visible":true},{"tag_id":31354,"tag_text":"summer camp","is_visible":true},{"tag_id":63,"tag_text":"summer program","is_visible":true},{"tag_id":169344,"tag_text":"summer math camp","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Does anyone know when the decisions for HCSSiM will come out this year?<br>\n<br>\nPlease post info if you received the decision letter.<br>\n<br>\n","category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","num_reports":0,"poll_id":0,"source":"","category_num_users":9,"category_num_topics":29049,"category_num_posts":471546,"num_views":5544,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848261":{"num_posts":18,"posts_data":[{"post_id":25244194,"topic_id":2848261,"poster_id":519212,"post_rendered":"Does anyone have any mathcounts mock competitions? preferably for nationals.","post_canonical":"Does anyone have any mathcounts mock competitions? preferably for nationals.","username":"Leo2020","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1652991462,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_519212.png?t=1619968852","num_posts":532,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848261,"comment_count":18,"num_deleted":1,"topic_title":"Mathcounts mocks","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25244194,"first_poster_id":519212,"first_post_time":1652991462,"first_poster_name":"Leo2020","last_post_time":1653073576,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_519212.png?t=1619968852","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_595281.png","last_post_id":25250606,"last_poster_id":595281,"last_poster_name":"heheman","last_update_time":1653073576,"category_id":5,"is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"tags":[{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Does anyone have any mathcounts mock competitions? preferably for nationals.","category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","num_reports":0,"poll_id":0,"source":"","category_num_users":9,"category_num_topics":29049,"category_num_posts":471546,"num_views":541,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848761":{"num_posts":3,"posts_data":[{"post_id":25250138,"topic_id":2848761,"poster_id":208295,"post_rendered":"Title says it all really. Planning on grinding over the summer for AIME qual after not doing the AMC 10. I know I should do the former tests and mocks and study through all the books, but one of my main questions is how much Alcumus per day should I do, and what study things are available outside mocks and Alcumus? I've taken all of the AoPS courses in the past up to Calc, so review should be a lot easier as I already know things.","post_canonical":"Title says it all really. Planning on grinding over the summer for AIME qual after not doing the AMC 10. I know I should do the former tests and mocks and study through all the books, but one of my main questions is how much Alcumus per day should I do, and what study things are available outside mocks and Alcumus? I've taken all of the AoPS courses in the past up to Calc, so review should be a lot easier as I already know things.","username":"moab33","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1653067852,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_208295.png?t=1629751514","num_posts":2123,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848761,"comment_count":3,"num_deleted":0,"topic_title":"How to study for AMC 10 effectively?","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25250138,"first_poster_id":208295,"first_post_time":1653067852,"first_poster_name":"moab33","last_post_time":1653073498,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_208295.png?t=1629751514","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_283749.png?t=1611261266","last_post_id":25250600,"last_poster_id":283749,"last_poster_name":"dolphin7","last_update_time":1653073498,"category_id":5,"is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"tags":[{"tag_id":69960,"tag_text":"study","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Title says it all really. Planning on grinding over the summer for AIME qual after not doing the AMC 10. I know I should do the former tests and mocks and study through all the books, but one of my main questions is how much Alcumus per day should I do, and what study things are available outside mocks and Alcumus? I've taken all of the AoPS courses in the past up to Calc, so review should be a lot easier as I already know things.","category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","num_reports":0,"poll_id":0,"source":"","category_num_users":9,"category_num_topics":29049,"category_num_posts":471546,"num_views":74,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848336":{"num_posts":4,"posts_data":[{"post_id":25245216,"topic_id":2848336,"poster_id":953304,"post_rendered":"How much does attending the ross summer camp improve chances for getting into mit primes?<br>\n<br>\nAlso, I heard that you have to know some linear algebra for the pset. Is hefferson's linear algebra textbook good enough for this? Also are there any other areas outside of a standard high school curriculum that should be learned before trying the pset?<br>\n<br>\nFinally, are the problems similar to competition problems, are more like research? For example, would they be more similar to the Ross psets, or more like olympiad problems?","post_canonical":"How much does attending the ross summer camp improve chances for getting into mit primes?\n\nAlso, I heard that you have to know some linear algebra for the pset. Is hefferson's linear algebra textbook good enough for this? Also are there any other areas outside of a standard high school curriculum that should be learned before trying the pset?\n\nFinally, are the problems similar to competition problems, are more like research? For example, would they be more similar to the Ross psets, or more like olympiad problems?","username":"math4life2023","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1652999786,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_953304.png","num_posts":2,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848336,"comment_count":4,"num_deleted":0,"topic_title":"Ross boost for mit primes?","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25245216,"first_poster_id":953304,"first_post_time":1652999786,"first_poster_name":"math4life2023","last_post_time":1653063788,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_953304.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_584480.png","last_post_id":25249758,"last_poster_id":584480,"last_poster_name":"CircleInvert","last_update_time":1653063788,"category_id":5,"is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"tags":[{"tag_id":84,"tag_text":"Ross Mathematics Program","is_visible":true},{"tag_id":135,"tag_text":"MIT","is_visible":true},{"tag_id":209,"tag_text":"college","is_visible":true},{"tag_id":290,"tag_text":"linear algebra","is_visible":true},{"tag_id":48,"tag_text":"geometry","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"How much does attending the ross summer camp improve chances for getting into mit primes?<br>\n<br>\nAlso, I heard that you have to know some linear algebra for the pset. Is hefferson's linear algebra textbook good enough for this? Also are there any other areas outside of a standard high school curriculum that should be learned before trying the pset?<br>\n<br>\nFinally, are the problems similar to competition problems, are more like research? For example, would they be more similar to the Ross psets, or more like olympiad problems?","category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","num_reports":0,"poll_id":0,"source":"","category_num_users":9,"category_num_topics":29049,"category_num_posts":471546,"num_views":470,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[6]={"2848792":{"num_posts":1,"posts_data":[{"post_id":25250634,"topic_id":2848792,"poster_id":167643,"post_rendered":"<b>VII.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.1 \/ J1.1<\/b> What is the parity of the number of ways to cut a rectangle into rectangles?<br>\n<br>\n<br>\n<b>1.2 \/ J1.2<\/b> In an isosceles triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > <span style=\"white-space:pre;\">(<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/9\/3c99bbe74cccdc92b249d376f13d9429d41a8b7a.png\" class=\"latex\" alt=\"$AB=AC$\" width=\"80\" height=\"13\" >)<\/span>. The circles <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > have different radii and lie outside the triangle. In this case, <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >.<\/span> Find the locus of the intersection of the common external tangents to these circles.<br>\n<br>\n<br>\n<b>1.5 \/ J1.5<\/b> There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)<br>\n<br>\n<br>\n<b>1.10 \/ J1.10<\/b> How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?<br>\n<br>\n<br>\n<b>2.1 \/ J2.1<\/b> Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.<br>\n<br>\n<br>\n<b>2.5 \/ J2.5<\/b> The plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > touches the sphere circumscribed around the tetrahedron <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >.<\/span> Prove that the lines of intersection of the planes of the faces <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/e\/6ce2cd123db24a7b8373999aea7a98927e3b3c81.png\" class=\"latex\" alt=\"$ACD$\" width=\"43\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/6\/a\/a6adfd3ad27a2dddeb0af7131d73c33c86991e93.png\" class=\"latex\" alt=\"$ABD$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > with the plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > divide it into six equal angles if and only if <img src=\"\/\/latex.artofproblemsolving.com\/0\/d\/6\/0d6daca6b8731859ee4ad0aaf1e473c1d920b8cb.png\" class=\"latex\" alt=\"$AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$\" style=\"vertical-align: -1px\" width=\"264\" height=\"14\" > .<br>\n<br>\n<br>\n<b>2.8 \/ J2.8<\/b> Is it possible to place a segment of length <img src=\"\/\/latex.artofproblemsolving.com\/4\/9\/4\/4944bcd1e5f1c6890c49160993314481c933217d.png\" class=\"latex\" alt=\"$1996$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" > on the checkered plane and choose a point <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > so that this segment does not touch nodes during any rotation around the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" >?<\/span><br>\n<br>\n<br>\n<b>2.9 \/ J2.9<\/b> The line intersects two concentric circles successively at the points <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/8\/a9874ea56cf8fec1bd246793224fbcfaad17ab3f.png\" class=\"latex\" alt=\"$A,B,C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > be the parallel chords of these circles, the points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > are the feet of the perpendiculars dropped from the points <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >,<\/span> respectively, on <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" >.<\/span> Prove that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/b\/5\/8b59ce9ac1bb33d8ddf80e77781727977923b0d0.png\" class=\"latex\" alt=\"$KF=ME$\" style=\"vertical-align: -1px\" width=\"91\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>3.2 \/ J3.3 <\/b> There are <img src=\"\/\/latex.artofproblemsolving.com\/8\/f\/1\/8f1e2533d6abdb11a9a1626c0240fab1a9200fc6.png\" class=\"latex\" alt=\"$n \\ge 3$\" style=\"vertical-align: -2px\" width=\"43\" height=\"15\" > points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?<br>\n<br>\n<br>\n<b>3.6<\/b> Points <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/2\/7\/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" > are the centers of the circumscribed and inscribed circles of an acute-angled triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is the foot of the altitude drawn from the vertex <img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > on the side <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >.<\/span> Prove that if the radius of the circumscribed circle is equal to the radius of the excircle tangent to the side <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > , then the points <img src=\"\/\/latex.artofproblemsolving.com\/0\/f\/f\/0fff95bb5e87cffa859314799ed3be4c25c1fe81.png\" class=\"latex\" alt=\"$O,I$\" style=\"vertical-align: -3px\" width=\"32\" height=\"16\" > and <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > lie on the same straight line.<br>\n<br>\n<br>\n<b>3.10<\/b> The sum of all plane angles of a convex <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-<\/span>hedron is equal to the sum of all its dihedral angles. Under what conditions is this possible?<br>\n<br>\n<br>\n<b>4.2<\/b> For each vertex of the tetrahedron, we construct a sphere passing through this vertex and the midpoints of the edges emerging from it. What is the largest number of the constructed 4 spheres that can touch the faces of the tetrahedron.<br>\n<br>\n<br>\n<b>4.4<\/b> On the sides <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/3\/0b31cd391bebc36897a8c10ea78e5d584980b049.png\" class=\"latex\" alt=\"$DA$\" width=\"29\" height=\"13\" > of the inscribed quadrilateral <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> points <img src=\"\/\/latex.artofproblemsolving.com\/7\/7\/5\/775865e36957a69a7a83638c624c6525448cbefb.png\" class=\"latex\" alt=\"$K,L,M,N$\" style=\"vertical-align: -3px\" width=\"87\" height=\"16\" > are taken, respectively, so that <img src=\"\/\/latex.artofproblemsolving.com\/c\/8\/b\/c8baa210c35a8d98d01f9f4d4c956c5a7bcd15ae.png\" class=\"latex\" alt=\"$KLMN$\" style=\"vertical-align: -1px\" width=\"67\" height=\"14\" > is a parallelogram. It is known that lines <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/c\/5\/bc56862b1fdb71204894da278e42ef75de73a40b.png\" class=\"latex\" alt=\"$AD$\" width=\"28\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/f\/3\/f\/f3fdf465274d91e623cc580002c0cc42fa72e36e.png\" class=\"latex\" alt=\"$KM$\" width=\"35\" height=\"12\" > pass through one point, lines <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/2\/7\/e\/27e525b776c49bbe7d0ae66cfc7ab8f58f40624e.png\" class=\"latex\" alt=\"$DC$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/e\/b\/beb5291b88ed134fe5f0f382c4301ba1686f2d77.png\" class=\"latex\" alt=\"$NL$\" style=\"vertical-align: -1px\" width=\"31\" height=\"14\" > also pass through one point. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/c\/8\/b\/c8baa210c35a8d98d01f9f4d4c956c5a7bcd15ae.png\" class=\"latex\" alt=\"$KLMN$\" style=\"vertical-align: -1px\" width=\"67\" height=\"14\" > is a rhombus.<br>\n<br>\n<br>\n<b>4.6 \/ J4.3<\/b> Given a "parallel ruler", with which you can draw a straight line through <img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/c\/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.<br>\n<br>\n<br>\n<b>4.9<\/b> Given <img src=\"\/\/latex.artofproblemsolving.com\/6\/0\/1\/601a7806cbfad68196c43a4665871f8c3186802e.png\" class=\"latex\" alt=\"$6$\" width=\"8\" height=\"12\" > points on the plane such that the distance between any two of them is at least <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" >.<\/span> Prove that among them there are two points whose distance between them is at least <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/e\/2\/ce27b68ffb7ffe98bb9caf4168a573b096205a0d.png\" class=\"latex\" alt=\"$\\sqrt{\\frac{5+\\sqrt5}{2}}$\" style=\"vertical-align: -16px\" width=\"78\" height=\"53\" >.<\/span><br>\n<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Junior Rounds have been posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c4h2848786p25250572\" class=\"bbcode_url\" target=\"_blank\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\" target=\"_blank\">here<\/a>.","post_canonical":"[b]VII.[\/b] [u]Rounds 1-4[\/u] \n\n\n[b]1.1 \/ J1.1[\/b] What is the parity of the number of ways to cut a rectangle into rectangles?\n\n\n[b]1.2 \/ J1.2[\/b] In an isosceles triangle $ABC$ ($AB=AC$). The circles $S_1$ and $S_2$ have different radii and lie outside the triangle. In this case, $S_1$ touches the line $AB$ at the point $A$, and $S_2$ touches the line $BC$ at the point $C$. Find the locus of the intersection of the common external tangents to these circles.\n\n\n[b]1.5 \/ J1.5[\/b] There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)\n\n\n[b]1.10 \/ J1.10[\/b] How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?\n\n\n[b]2.1 \/ J2.1[\/b] Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.\n\n\n[b]2.5 \/ J2.5[\/b] The plane $\\alpha$ touches the sphere circumscribed around the tetrahedron $ABCD$ at the point $A$. Prove that the lines of intersection of the planes of the faces $ABC$, $ACD$ and $ABD$ with the plane $\\alpha$ divide it into six equal angles if and only if $AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$ .\n\n\n[b]2.8 \/ J2.8[\/b] Is it possible to place a segment of length $1996$ on the checkered plane and choose a point $O$ so that this segment does not touch nodes during any rotation around the point $O$?\n\n\n[b]2.9 \/ J2.9[\/b] The line intersects two concentric circles successively at the points $A,B,C$ and $D$. Let $AE$ and $BF$ be the parallel chords of these circles, the points $K$ and $M$ are the feet of the perpendiculars dropped from the points $C$ and $D$, respectively, on $BF$ and $AE$. Prove that $KF=ME$.\n\n\n[b]3.2 \/ J3.3 [\/b] There are $n \\ge 3$ points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?\n\n\n[b]3.6[\/b] Points $O$ and $I$ are the centers of the circumscribed and inscribed circles of an acute-angled triangle $ABC$, $D$ is the foot of the altitude drawn from the vertex $A$ on the side $BC$. Prove that if the radius of the circumscribed circle is equal to the radius of the excircle tangent to the side $BC$ , then the points $O,I$ and $D$ lie on the same straight line.\n\n\n[b]3.10[\/b] The sum of all plane angles of a convex $n$-hedron is equal to the sum of all its dihedral angles. Under what conditions is this possible?\n\n\n[b]4.2[\/b] For each vertex of the tetrahedron, we construct a sphere passing through this vertex and the midpoints of the edges emerging from it. What is the largest number of the constructed 4 spheres that can touch the faces of the tetrahedron.\n\n\n[b]4.4[\/b] On the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$, points $K,L,M,N$ are taken, respectively, so that $KLMN$ is a parallelogram. It is known that lines $AD$, $BC$ and $KM$ pass through one point, lines $AB$, $DC$ and $NL$ also pass through one point. Prove that $KLMN$ is a rhombus.\n\n\n[b]4.6 \/ J4.3[\/b] Given a \"parallel ruler\", with which you can draw a straight line through $2$ points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.\n\n\n[b]4.9[\/b] Given $6$ points on the plane such that the distance between any two of them is at least $1$. Prove that among them there are two points whose distance between them is at least $\\sqrt{\\frac{5+\\sqrt5}{2}}$.\n\n\n\n\nPS. You should use hide for answers. Junior Rounds have been posted [url=https:\/\/artofproblemsolving.com\/community\/c4h2848786p25250572]here[\/url]. Collected [url=https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry]here[\/url].","username":"parmenides51","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1653073912,"num_edits":1,"post_format":"bbcode","last_edit_time":1653074047,"last_editor_username":"parmenides51","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","num_posts":18899,"editable":false,"deletable":true,"show_from_start":true,"show_from_end":true}],"topic_id":2848792,"comment_count":1,"num_deleted":0,"topic_title":"Geo from VII Russian Festival of Young Mathematicians 1996 Rounds","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25250634,"first_poster_id":167643,"first_post_time":1653073912,"first_poster_name":"parmenides51","last_post_time":1653073912,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_167643.png?t=1472410314","last_post_id":25250634,"last_poster_id":167643,"last_poster_name":"parmenides51","last_update_time":1653074047,"category_id":6,"is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":146,"tag_text":"3D geometry","is_visible":true},{"tag_id":1600924,"tag_text":"RFYM","is_visible":true},{"tag_id":41772,"tag_text":"combinatorial geometry","is_visible":true},{"tag_id":267,"tag_text":"geometric inequality","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"<b>VII.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.1 \/ J1.1<\/b> What is the parity of the number of ways to cut a rectangle into rectangles?<br>\n<br>\n<br>\n<b>1.2 \/ J1.2<\/b> In an isosceles triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > <span style=\"white-space:pre;\">(<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/9\/3c99bbe74cccdc92b249d376f13d9429d41a8b7a.png\" class=\"latex\" alt=\"$AB=AC$\" width=\"80\" height=\"13\" >)<\/span>. The circles <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > have different radii and lie outside the triangle. In this case, <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/0\/b10d5ee88ffc7d59f9c282b366953fb4fa4051f6.png\" class=\"latex\" alt=\"$S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/7\/f\/4\/7f4caea56d945d0b6e49701b42ce06e0fb92e663.png\" class=\"latex\" alt=\"$S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > touches the line <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >.<\/span> Find the locus of the intersection of the common external tangents to these circles.<br>\n<br>\n<br>\n<b>1.5 \/ J1.5<\/b> There is a triangular pyramid on the table. Its edges are divided into equal parts and straight lines are drawn through the division points on each side face, parallel to the sides of the corresponding triangle and forming triangular grids there. At the initial moment, the magnetic chip is located at the top of the tetrahedron (not lying on the table). Two people take turns moving this piece over the nodes of the grid. In one move, it is allowed to move a chip along the grid to an adjacent node lying either in the same horizontal plane or lower. It is forbidden to put a chip in those nodes where it has already visited. The one who cannot make the next move loses. Who will win with the right play: the beginner or his opponent? (All edges and nodes lying on them belong to the grid)<br>\n<br>\n<br>\n<b>1.10 \/ J1.10<\/b> How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?<br>\n<br>\n<br>\n<b>2.1 \/ J2.1<\/b> Prove that for any convex polygon there exist two parallelograms that are homothetic with a factor of two and such that one of them contains the given polygon and the other is contained in it.<br>\n<br>\n<br>\n<b>2.5 \/ J2.5<\/b> The plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > touches the sphere circumscribed around the tetrahedron <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > at the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >.<\/span> Prove that the lines of intersection of the planes of the faces <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/e\/6ce2cd123db24a7b8373999aea7a98927e3b3c81.png\" class=\"latex\" alt=\"$ACD$\" width=\"43\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/6\/a\/a6adfd3ad27a2dddeb0af7131d73c33c86991e93.png\" class=\"latex\" alt=\"$ABD$\" style=\"vertical-align: -1px\" width=\"46\" height=\"14\" > with the plane <img src=\"\/\/latex.artofproblemsolving.com\/1\/0\/f\/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > divide it into six equal angles if and only if <img src=\"\/\/latex.artofproblemsolving.com\/0\/d\/6\/0d6daca6b8731859ee4ad0aaf1e473c1d920b8cb.png\" class=\"latex\" alt=\"$AB \\cdot CD = AC \\cdot BD = AD \\cdot BC$\" style=\"vertical-align: -1px\" width=\"264\" height=\"14\" > .<br>\n<br>\n<br>\n<b>2.8 \/ J2.8<\/b> Is it possible to place a segment of length <img src=\"\/\/latex.artofproblemsolving.com\/4\/9\/4\/4944bcd1e5f1c6890c49160993314481c933217d.png\" class=\"latex\" alt=\"$1996$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" > on the checkered plane and choose a point <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > so that this segment does not touch nodes during any rotation around the point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" >?<\/span><br>\n<br>\n<br>\n<b>2.9 \/ J2.9<\/b> The line intersects two concentric circles successively at the points <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/8\/a9874ea56cf8fec1bd246793224fbcfaad17ab3f.png\" class=\"latex\" alt=\"$A,B,C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > be the parallel chords of these circles, the points <img src=\"\/\/latex.artofproblemsolving.com\/d\/f\/b\/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" style=\"vertical-align: -1px\" width=\"19\" height=\"14\" > and <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > are the feet of the perpendiculars dropped from the points <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >,<\/span> respectively, on <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65e4f2d5cc59e5ffbed3734130e8c45b643adc91.png\" class=\"latex\" alt=\"$BF$\" width=\"28\" height=\"12\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/6\/9a65c05bd85d2609bea5db2b109ed5f556463511.png\" class=\"latex\" alt=\"$AE$\" width=\"27\" height=\"13\" >.<\/span> Prove that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/b\/5\/8b59ce9ac1bb33d8ddf80e77781727977923b0d0.png\" class=\"latex\" alt=\"$KF=ME$\" style=\"vertical-align: -1px\" width=\"91\" height=\"14\" >.<\/span><br>\n<br>\n<br>\n<b>3.2 \/ J3.3 <\/b> There are <img src=\"\/\/latex.artofproblemsolving.com\/8\/f\/1\/8f1e2533d6abdb11a9a1626c0240fab1a9200fc6.png\" class=\"latex\" alt=\"$n \\ge 3$\" style=\"vertical-align: -2px\" width=\"43\" height=\"15\" > points marked on the plane, no three of which lie on the same straight line. Two people play the following game: take turns connecting pairs of points with segments that have not been drawn. The one that loses, after the course of which a triangle is formed with vertices at the marked points. Who wins if they both paly optimal?<br>\n<br>\n<br>\n<b>3.6<\/b> Points <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/d\/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/2\/7\/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" > are the centers of the circumscribed and inscribed circles of an acute-angled triangle <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is the foot of the altitude drawn from the vertex <img src=\"\/\/latex.artofproblemsolving.com\/0\/1\/9\/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > on the side <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >.<\/span> Prove that if the radius of the circumscribed circle is equal to the radius of the excircle tangent to the side <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > , then the points <img src=\"\/\/latex.artofproblemsolving.com\/0\/f\/f\/0fff95bb5e87cffa859314799ed3be4c25c1fe81.png\" class=\"latex\" alt=\"$O,I$\" style=\"vertical-align: -3px\" width=\"32\" height=\"16\" > and <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > lie on the same straight line.<br>\n<br>\n<br>\n<b>3.10<\/b> The sum of all plane angles of a convex <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-<\/span>hedron is equal to the sum of all its dihedral angles. Under what conditions is this possible?<br>\n<br>\n<br>\n<b>4.2<\/b> For each vertex of the tetrahedron, we construct a sphere passing through this vertex and the midpoints of the edges emerging from it. What is the largest number of the constructed 4 spheres that can touch the faces of the tetrahedron.<br>\n<br>\n<br>\n<b>4.4<\/b> On the sides <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/3\/0b31cd391bebc36897a8c10ea78e5d584980b049.png\" class=\"latex\" alt=\"$DA$\" width=\"29\" height=\"13\" > of the inscribed quadrilateral <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >,<\/span> points <img src=\"\/\/latex.artofproblemsolving.com\/7\/7\/5\/775865e36957a69a7a83638c624c6525448cbefb.png\" class=\"latex\" alt=\"$K,L,M,N$\" style=\"vertical-align: -3px\" width=\"87\" height=\"16\" > are taken, respectively, so that <img src=\"\/\/latex.artofproblemsolving.com\/c\/8\/b\/c8baa210c35a8d98d01f9f4d4c956c5a7bcd15ae.png\" class=\"latex\" alt=\"$KLMN$\" style=\"vertical-align: -1px\" width=\"67\" height=\"14\" > is a parallelogram. It is known that lines <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/c\/5\/bc56862b1fdb71204894da278e42ef75de73a40b.png\" class=\"latex\" alt=\"$AD$\" width=\"28\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/f\/3\/f\/f3fdf465274d91e623cc580002c0cc42fa72e36e.png\" class=\"latex\" alt=\"$KM$\" width=\"35\" height=\"12\" > pass through one point, lines <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/2\/7\/e\/27e525b776c49bbe7d0ae66cfc7ab8f58f40624e.png\" class=\"latex\" alt=\"$DC$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/e\/b\/beb5291b88ed134fe5f0f382c4301ba1686f2d77.png\" class=\"latex\" alt=\"$NL$\" style=\"vertical-align: -1px\" width=\"31\" height=\"14\" > also pass through one point. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/c\/8\/b\/c8baa210c35a8d98d01f9f4d4c956c5a7bcd15ae.png\" class=\"latex\" alt=\"$KLMN$\" style=\"vertical-align: -1px\" width=\"67\" height=\"14\" > is a rhombus.<br>\n<br>\n<br>\n<b>4.6 \/ J4.3<\/b> Given a "parallel ruler", with which you can draw a straight line through <img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/c\/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > points, and also determine whether these two lines are parallel or not. Using a parallel ruler and scissors, cut out a triangle of maximum area from a given non-convex quadrilateral.<br>\n<br>\n<br>\n<b>4.9<\/b> Given <img src=\"\/\/latex.artofproblemsolving.com\/6\/0\/1\/601a7806cbfad68196c43a4665871f8c3186802e.png\" class=\"latex\" alt=\"$6$\" width=\"8\" height=\"12\" > points on the plane such that the distance between any two of them is at least <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" >.<\/span> Prove that among them there are two points whose distance between them is at least <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/e\/2\/ce27b68ffb7ffe98bb9caf4168a573b096205a0d.png\" class=\"latex\" alt=\"$\\sqrt{\\frac{5+\\sqrt5}{2}}$\" style=\"vertical-align: -16px\" width=\"78\" height=\"53\" >.<\/span><br>\n<br>\n<br>\n<br>\n<br>\nPS. You should use hide for answers. Junior Rounds have been posted <a href=\"https:\/\/artofproblemsolving.com\/community\/c4h2848786p25250572\" class=\"bbcode_url\">here<\/a>. Collected <a href=\"https:\/\/artofproblemsolving.com\/community\/c3032551_rfym__geometry\" class=\"bbcode_url\">here<\/a>.","category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","num_reports":0,"poll_id":0,"source":"VII Russian Festival of Young Mathematicians 1996 https:\/\/artofproblemsolving.com\/community\/c3032551_","category_num_users":18,"category_num_topics":275646,"category_num_posts":1201309,"num_views":2,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":true,"in_feed":true,"is_watched":true},"2764594":{"num_posts":4,"posts_data":[{"post_id":24200806,"topic_id":2764594,"poster_id":827681,"post_rendered":"If there are integers <img src=\"\/\/latex.artofproblemsolving.com\/a\/5\/b\/a5b29b358e825defa3e11b7d903d43ec31e5909a.png\" class=\"latex\" alt=\"$a,b,c$\" style=\"vertical-align: -3px\" width=\"41\" height=\"16\" > such that <img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/9\/9b9ad12c610cc006bb8486a46c73129e6dbe7b18.png\" class=\"latex\" alt=\"$a^2+b^2+c^2-ab-bc-ca$\" style=\"vertical-align: -1px\" width=\"209\" height=\"16\" > is divisible by a prime <img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/f\/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/7\/d\/a7da8257eab459bb3852b3d888ba563cc6aa00ed.png\" class=\"latex\" alt=\"$\\text{gcd}(p,\\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$\" style=\"vertical-align: -17px\" width=\"305\" height=\"44\" >,<\/span> then prove that there are integers <img src=\"\/\/latex.artofproblemsolving.com\/a\/c\/c\/accc80fdf164cef264f56a82b6f9f6add320fe05.png\" class=\"latex\" alt=\"$x,y,z$\" style=\"vertical-align: -3px\" width=\"44\" height=\"11\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/1\/0b18d2b8cb98b666e2b3263d641d56d9e982236b.png\" class=\"latex\" alt=\"$p=x^2+y^2+z^2-xy-yz-zx$\" style=\"vertical-align: -4px\" width=\"256\" height=\"20\" >.<\/span>","post_canonical":" If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\\text{gcd}(p,\\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.","username":"Project_Donkey_into_M4","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1643019375,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_827681.png","num_posts":102,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2764594,"comment_count":4,"num_deleted":2,"topic_title":"Another strange representation of a prime","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":24200806,"first_poster_id":827681,"first_post_time":1643019375,"first_poster_name":"Project_Donkey_into_M4","last_post_time":1653073479,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_827681.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_606915.jpg?t=1641491987","last_post_id":25250598,"last_poster_id":606915,"last_poster_name":"nathantareep","last_update_time":1653073479,"category_id":6,"is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"tags":[{"tag_id":177,"tag_text":"number theory","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":" If there are integers <img src=\"\/\/latex.artofproblemsolving.com\/a\/5\/b\/a5b29b358e825defa3e11b7d903d43ec31e5909a.png\" class=\"latex\" alt=\"$a,b,c$\" style=\"vertical-align: -3px\" width=\"41\" height=\"16\" > such that <img src=\"\/\/latex.artofproblemsolving.com\/9\/b\/9\/9b9ad12c610cc006bb8486a46c73129e6dbe7b18.png\" class=\"latex\" alt=\"$a^2+b^2+c^2-ab-bc-ca$\" style=\"vertical-align: -1px\" width=\"209\" height=\"16\" > is divisible by a prime <img src=\"\/\/latex.artofproblemsolving.com\/3\/6\/f\/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/7\/d\/a7da8257eab459bb3852b3d888ba563cc6aa00ed.png\" class=\"latex\" alt=\"$\\text{gcd}(p,\\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$\" style=\"vertical-align: -17px\" width=\"305\" height=\"44\" >,<\/span> then prove that there are integers <img src=\"\/\/latex.artofproblemsolving.com\/a\/c\/c\/accc80fdf164cef264f56a82b6f9f6add320fe05.png\" class=\"latex\" alt=\"$x,y,z$\" style=\"vertical-align: -3px\" width=\"44\" height=\"11\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/1\/0b18d2b8cb98b666e2b3263d641d56d9e982236b.png\" class=\"latex\" alt=\"$p=x^2+y^2+z^2-xy-yz-zx$\" style=\"vertical-align: -4px\" width=\"256\" height=\"20\" >.<\/span>","category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","num_reports":0,"poll_id":0,"source":"STEMS 2022 Math Cat A P3\/Cat B P4\/Cat C P1","category_num_users":18,"category_num_topics":275646,"category_num_posts":1201309,"num_views":293,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2835390":{"num_posts":11,"posts_data":[{"post_id":25099531,"topic_id":2835390,"poster_id":621197,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > be a convex quadrilateral, the incenters of <img src=\"\/\/latex.artofproblemsolving.com\/8\/c\/3\/8c3a2d2224f7d163b46d702132425d47828bf538.png\" class=\"latex\" alt=\"$\\triangle ABC$\" width=\"58\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/8\/e\/b8eabc87afe4c11a47e4aa27628d5517af4866b9.png\" class=\"latex\" alt=\"$\\triangle ADC$\" width=\"59\" height=\"13\" > are <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/0\/a\/b0a151a192d1846a8561e6dd87f893af7efbb262.png\" class=\"latex\" alt=\"$I,J$\" style=\"vertical-align: -3px\" width=\"29\" height=\"16\" >,<\/span> respectively. It is known that <img src=\"\/\/latex.artofproblemsolving.com\/8\/4\/d\/84d67e0f07ea443c3a18b8221c7f55d31f810b0e.png\" class=\"latex\" alt=\"$AC,BD,IJ$\" style=\"vertical-align: -3px\" width=\"95\" height=\"16\" > concurrent at a point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >.<\/span> The line perpendicular to <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > through <img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > intersects with the outer angle bisector of <img src=\"\/\/latex.artofproblemsolving.com\/c\/b\/5\/cb55dc9c50a80e1d9eb23fe53e45f464c0ac755e.png\" class=\"latex\" alt=\"$\\angle BAD$\" width=\"56\" height=\"13\" > and the outer angle bisector <img src=\"\/\/latex.artofproblemsolving.com\/2\/3\/3\/23386faeefe37629243fa264877b288499db5048.png\" class=\"latex\" alt=\"$\\angle BCD$\" width=\"57\" height=\"12\" > at <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/2\/5\/0\/250f765e0f0b3842f9d095dc0a3569905d344ce8.png\" class=\"latex\" alt=\"$E,F$\" style=\"vertical-align: -3px\" width=\"36\" height=\"16\" >,<\/span> respectively. Show that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/e\/f\/eef01363603050cadb084777489b45552600fa2b.png\" class=\"latex\" alt=\"$PE=PF$\" width=\"81\" height=\"12\" >.<\/span>","post_canonical":"Let $ABCD$ be a convex quadrilateral, the incenters of $\\triangle ABC$ and $\\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\\angle BAD$ and the outer angle bisector $\\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.","username":"JustPostChinaTST","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1651333706,"num_edits":3,"post_format":"bbcode","last_edit_time":1651507883,"last_editor_username":"JustPostChinaTST","last_edit_reason":"typos","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_621197.png","num_posts":38,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2835390,"comment_count":11,"num_deleted":0,"topic_title":"Concurrency in a quadrilateral implies equal length","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25099531,"first_poster_id":621197,"first_post_time":1651333706,"first_poster_name":"JustPostChinaTST","last_post_time":1653072337,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_621197.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_713747.jpg?t=1648118822","last_post_id":25250508,"last_poster_id":713747,"last_poster_name":"GuvercinciHoca","last_update_time":1653072337,"category_id":6,"is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":120,"tag_text":"incenter","is_visible":true},{"tag_id":124,"tag_text":"angle bisector","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > be a convex quadrilateral, the incenters of <img src=\"\/\/latex.artofproblemsolving.com\/8\/c\/3\/8c3a2d2224f7d163b46d702132425d47828bf538.png\" class=\"latex\" alt=\"$\\triangle ABC$\" width=\"58\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/8\/e\/b8eabc87afe4c11a47e4aa27628d5517af4866b9.png\" class=\"latex\" alt=\"$\\triangle ADC$\" width=\"59\" height=\"13\" > are <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/0\/a\/b0a151a192d1846a8561e6dd87f893af7efbb262.png\" class=\"latex\" alt=\"$I,J$\" style=\"vertical-align: -3px\" width=\"29\" height=\"16\" >,<\/span> respectively. It is known that <img src=\"\/\/latex.artofproblemsolving.com\/8\/4\/d\/84d67e0f07ea443c3a18b8221c7f55d31f810b0e.png\" class=\"latex\" alt=\"$AC,BD,IJ$\" style=\"vertical-align: -3px\" width=\"95\" height=\"16\" > concurrent at a point <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >.<\/span> The line perpendicular to <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > through <img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > intersects with the outer angle bisector of <img src=\"\/\/latex.artofproblemsolving.com\/c\/b\/5\/cb55dc9c50a80e1d9eb23fe53e45f464c0ac755e.png\" class=\"latex\" alt=\"$\\angle BAD$\" width=\"56\" height=\"13\" > and the outer angle bisector <img src=\"\/\/latex.artofproblemsolving.com\/2\/3\/3\/23386faeefe37629243fa264877b288499db5048.png\" class=\"latex\" alt=\"$\\angle BCD$\" width=\"57\" height=\"12\" > at <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/2\/5\/0\/250f765e0f0b3842f9d095dc0a3569905d344ce8.png\" class=\"latex\" alt=\"$E,F$\" style=\"vertical-align: -3px\" width=\"36\" height=\"16\" >,<\/span> respectively. Show that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/e\/f\/eef01363603050cadb084777489b45552600fa2b.png\" class=\"latex\" alt=\"$PE=PF$\" width=\"81\" height=\"12\" >.<\/span>","category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","num_reports":0,"poll_id":0,"source":"2022 China TST, Test 4 P2","category_num_users":18,"category_num_topics":275646,"category_num_posts":1201309,"num_views":1190,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"1374313":{"num_posts":7,"posts_data":[{"post_id":7587802,"topic_id":1374313,"poster_id":232609,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > be a cyclic quadrilateral. The diagonals <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/c\/0\/2\/c021ecae123b8c58ceff108d560db0544f0ccb8e.png\" class=\"latex\" alt=\"$DA $\" width=\"29\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/3\/e\/3\/3e3a4ec0df571e719bd3f1d7e00bf71fc55ba2b9.png\" class=\"latex\" alt=\"$CB$\" width=\"28\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/8\/6\/9866e3a998d628ba0941eb4fea0666ac391d149a.png\" class=\"latex\" alt=\"$Q$\" style=\"vertical-align: -3px\" width=\"13\" height=\"16\" >.<\/span> Suppose <img src=\"\/\/latex.artofproblemsolving.com\/7\/a\/b\/7ab17967324fc2b12349c0610c6c256f4c829162.png\" class=\"latex\" alt=\"$PQ$\" style=\"vertical-align: -3px\" width=\"27\" height=\"16\" > is perpendicular to <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > be the midpoint of <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >.<\/span> Prove that <img src=\"\/\/latex.artofproblemsolving.com\/0\/e\/d\/0edebe902653b343eee6f0a6a93a09f80884504e.png\" class=\"latex\" alt=\"$PE$\" width=\"28\" height=\"12\" > is perpendicular to <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >.<\/span>","post_canonical":"Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA $ and $CB$ meet at $Q$. Suppose $PQ$ is perpendicular to $AC$. Let $E$ be the midpoint of $AB$. Prove that $PE$ is perpendicular to $BC$.","username":"sualehasif996","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1485630962,"num_edits":2,"post_format":"bbcode","last_edit_time":1485631009,"last_editor_username":"sualehasif996","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_232609.png","num_posts":110,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":1374313,"comment_count":7,"num_deleted":0,"topic_title":"Perpendicularity in Cyclic Quad.","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":7587802,"first_poster_id":232609,"first_post_time":1485630962,"first_poster_name":"sualehasif996","last_post_time":1653071847,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_232609.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_421983.png?t=1559877586","last_post_id":25250472,"last_poster_id":421983,"last_poster_name":"TheCoolDinosuar","last_update_time":1653071847,"category_id":6,"is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"tags":[{"tag_id":210,"tag_text":"cyclic quadrilateral","is_visible":true},{"tag_id":129996,"tag_text":"perpendicular lines","is_visible":true},{"tag_id":48,"tag_text":"geometry","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/9\/e\/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > be a cyclic quadrilateral. The diagonals <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/b\/4\/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/c\/0\/2\/c021ecae123b8c58ceff108d560db0544f0ccb8e.png\" class=\"latex\" alt=\"$DA $\" width=\"29\" height=\"13\" > and <img src=\"\/\/latex.artofproblemsolving.com\/3\/e\/3\/3e3a4ec0df571e719bd3f1d7e00bf71fc55ba2b9.png\" class=\"latex\" alt=\"$CB$\" width=\"28\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/8\/6\/9866e3a998d628ba0941eb4fea0666ac391d149a.png\" class=\"latex\" alt=\"$Q$\" style=\"vertical-align: -3px\" width=\"13\" height=\"16\" >.<\/span> Suppose <img src=\"\/\/latex.artofproblemsolving.com\/7\/a\/b\/7ab17967324fc2b12349c0610c6c256f4c829162.png\" class=\"latex\" alt=\"$PQ$\" style=\"vertical-align: -3px\" width=\"27\" height=\"16\" > is perpendicular to <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > be the midpoint of <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >.<\/span> Prove that <img src=\"\/\/latex.artofproblemsolving.com\/0\/e\/d\/0edebe902653b343eee6f0a6a93a09f80884504e.png\" class=\"latex\" alt=\"$PE$\" width=\"28\" height=\"12\" > is perpendicular to <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/c\/5\/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >.<\/span>","category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","num_reports":0,"poll_id":0,"source":"Pakistan TST(2) 2017.P1","category_num_users":18,"category_num_topics":275646,"category_num_posts":1201309,"num_views":1967,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[7]={"2847484":{"num_posts":2,"posts_data":[{"post_id":25236311,"topic_id":2847484,"poster_id":407889,"post_rendered":"Given <b>u<\/b> = <img src=\"\/\/latex.artofproblemsolving.com\/e\/4\/1\/e4162d79742879fcd196374e6e48a3f896087872.png\" class=\"latex\" alt=\"$(u, v)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > with <img src=\"\/\/latex.artofproblemsolving.com\/5\/4\/a\/54a27443b8529381ce18428ff378affaf9af3813.png\" class=\"latex\" alt=\"$u= (e^x + 3x^2y)$\" style=\"vertical-align: -5px\" width=\"126\" height=\"21\" > and <img src=\"\/\/latex.artofproblemsolving.com\/c\/9\/8\/c985a4b0346e01be6f9a06e95acbfdbc75a55fed.png\" class=\"latex\" alt=\"$v= (e^{-y} + x^3 -4y^3)$\" style=\"vertical-align: -5px\" width=\"165\" height=\"21\" > and the circle <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > with radius <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/d\/0bde359170829f653d087534939847c27a92cd2a.png\" class=\"latex\" alt=\"$r = 1$\" style=\"vertical-align: 0px\" width=\"41\" height=\"12\" > and center at the origin.<br>\n<br>\nEvaluate the integral of <b>u<\/b>. d<b>r<\/b> <img src=\"\/\/latex.artofproblemsolving.com\/e\/1\/3\/e13ab851ecd5e9bfd436e89a608fdacd3cf2a846.png\" class=\"latex\" alt=\"$ = u$\" style=\"vertical-align: -1px\" width=\"31\" height=\"10\" > <img src=\"\/\/latex.artofproblemsolving.com\/9\/5\/3\/953640113c8002037caa26af92570f9cb7d5773c.png\" class=\"latex\" alt=\"$dx$\" width=\"19\" height=\"12\" > + <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/f\/a9f23bf124b6b2b2a993eb313c72e678664ac74a.png\" class=\"latex\" alt=\"$v$\" width=\"9\" height=\"8\" > <img src=\"\/\/latex.artofproblemsolving.com\/5\/a\/6\/5a687ff25b068f70b5def1fdb4f1c49344997391.png\" class=\"latex\" alt=\"$dy$\" style=\"vertical-align: -3px\" width=\"18\" height=\"16\" > on the circle from the point <img src=\"\/\/latex.artofproblemsolving.com\/7\/8\/6\/786c4ed8fdce54fb548034a4ff329c3c7dc0b8b4.png\" class=\"latex\" alt=\"$A : (1, 0)$\" style=\"vertical-align: -5px\" width=\"71\" height=\"20\" > to the point <img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/c\/d6c84e1264b804b33f9a4e3b7fb8db0bbb58f29b.png\" class=\"latex\" alt=\"$B: (0, 1).$\" style=\"vertical-align: -5px\" width=\"77\" height=\"20\" >","post_canonical":"Given [b]u[\/b] = $(u, v)$ with $u= (e^x + 3x^2y)$ and $v= (e^{-y} + x^3 -4y^3)$ and the circle $C$ with radius $r = 1$ and center at the origin.\n\nEvaluate the integral of [b]u[\/b]. d[b]r[\/b] $ = u$ $dx$ + $v$ $dy$ on the circle from the point $A : (1, 0)$ to the point $B: (0, 1).$","username":"DurdonTyler","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1652898282,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_407889.png","num_posts":93,"editable":false,"deletable":true,"show_from_start":true,"show_from_end":false}],"topic_id":2847484,"comment_count":2,"num_deleted":0,"topic_title":"evaluate int on the circle from a to b","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25236311,"first_poster_id":407889,"first_post_time":1652898282,"first_poster_name":"DurdonTyler","last_post_time":1653071393,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_407889.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_407889.png","last_post_id":25250440,"last_poster_id":407889,"last_poster_name":"DurdonTyler","last_update_time":1653071393,"category_id":7,"is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"tags":[{"tag_id":116,"tag_text":"calculus","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Given <b>u<\/b> = <img src=\"\/\/latex.artofproblemsolving.com\/e\/4\/1\/e4162d79742879fcd196374e6e48a3f896087872.png\" class=\"latex\" alt=\"$(u, v)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > with <img src=\"\/\/latex.artofproblemsolving.com\/5\/4\/a\/54a27443b8529381ce18428ff378affaf9af3813.png\" class=\"latex\" alt=\"$u= (e^x + 3x^2y)$\" style=\"vertical-align: -5px\" width=\"126\" height=\"21\" > and <img src=\"\/\/latex.artofproblemsolving.com\/c\/9\/8\/c985a4b0346e01be6f9a06e95acbfdbc75a55fed.png\" class=\"latex\" alt=\"$v= (e^{-y} + x^3 -4y^3)$\" style=\"vertical-align: -5px\" width=\"165\" height=\"21\" > and the circle <img src=\"\/\/latex.artofproblemsolving.com\/c\/3\/3\/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > with radius <img src=\"\/\/latex.artofproblemsolving.com\/0\/b\/d\/0bde359170829f653d087534939847c27a92cd2a.png\" class=\"latex\" alt=\"$r = 1$\" style=\"vertical-align: 0px\" width=\"41\" height=\"12\" > and center at the origin.<br>\n<br>\nEvaluate the integral of <b>u<\/b>. d<b>r<\/b> <img src=\"\/\/latex.artofproblemsolving.com\/e\/1\/3\/e13ab851ecd5e9bfd436e89a608fdacd3cf2a846.png\" class=\"latex\" alt=\"$ = u$\" style=\"vertical-align: -1px\" width=\"31\" height=\"10\" > <img src=\"\/\/latex.artofproblemsolving.com\/9\/5\/3\/953640113c8002037caa26af92570f9cb7d5773c.png\" class=\"latex\" alt=\"$dx$\" width=\"19\" height=\"12\" > + <img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/f\/a9f23bf124b6b2b2a993eb313c72e678664ac74a.png\" class=\"latex\" alt=\"$v$\" width=\"9\" height=\"8\" > <img src=\"\/\/latex.artofproblemsolving.com\/5\/a\/6\/5a687ff25b068f70b5def1fdb4f1c49344997391.png\" class=\"latex\" alt=\"$dy$\" style=\"vertical-align: -3px\" width=\"18\" height=\"16\" > on the circle from the point <img src=\"\/\/latex.artofproblemsolving.com\/7\/8\/6\/786c4ed8fdce54fb548034a4ff329c3c7dc0b8b4.png\" class=\"latex\" alt=\"$A : (1, 0)$\" style=\"vertical-align: -5px\" width=\"71\" height=\"20\" > to the point <img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/c\/d6c84e1264b804b33f9a4e3b7fb8db0bbb58f29b.png\" class=\"latex\" alt=\"$B: (0, 1).$\" style=\"vertical-align: -5px\" width=\"77\" height=\"20\" >","category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","num_reports":0,"poll_id":0,"source":"","category_num_users":5,"category_num_topics":102698,"category_num_posts":402685,"num_views":51,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2846853":{"num_posts":2,"posts_data":[{"post_id":25229502,"topic_id":2846853,"poster_id":355854,"post_rendered":"Show that the power series for the function<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/d\/8\/1\/d81856a3a61e5784dcfe52d905697a1a349edada.png\" class=\"latexcenter\" alt=\"$$e^{ax} \\cos bx,$$\" width=\"76\" height=\"16\" >where <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/f\/8\/bf8da8bc169153bc137eb64cefeafc6283c91cbb.png\" class=\"latex\" alt=\"$a,b >0$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" >,<\/span> has either no zero coefficients or infinitely many zero coefficients.","post_canonical":"Show that the power series for the function\n$$e^{ax} \\cos bx,$$\nwhere $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.\n","username":"sqrtX","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1652819785,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_355854.png","num_posts":344,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2846853,"comment_count":2,"num_deleted":0,"topic_title":"Putnam 1970 A1","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25229502,"first_poster_id":355854,"first_post_time":1652819785,"first_poster_name":"sqrtX","last_post_time":1653066635,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_355854.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_574587.png","last_post_id":25250049,"last_poster_id":574587,"last_poster_name":"maxjw91","last_update_time":1653066635,"category_id":7,"is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"tags":[{"tag_id":441,"tag_text":"Putnam","is_visible":true},{"tag_id":298,"tag_text":"function","is_visible":true},{"tag_id":194,"tag_text":"trigonometry","is_visible":true},{"tag_id":96815,"tag_text":"power series","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Show that the power series for the function<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/d\/8\/1\/d81856a3a61e5784dcfe52d905697a1a349edada.png\" class=\"latexcenter\" alt=\"$$e^{ax} \\cos bx,$$\" width=\"76\" height=\"16\" >where <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/f\/8\/bf8da8bc169153bc137eb64cefeafc6283c91cbb.png\" class=\"latex\" alt=\"$a,b >0$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" >,<\/span> has either no zero coefficients or infinitely many zero coefficients.<br>\n","category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","num_reports":0,"poll_id":0,"source":"Putnam 1970","category_num_users":5,"category_num_topics":102698,"category_num_posts":402685,"num_views":78,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2822357":{"num_posts":3,"posts_data":[{"post_id":24942736,"topic_id":2822357,"poster_id":142747,"post_rendered":"Reduce each equation<span style=\"white-space:pre;\">,<img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" ><\/span> through <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/d\/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" >,<\/span>to standard form. Then find the coordinates of the center ,the vertices , and the foci.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/3\/4\/a\/34a1b2b56eef892795b093bab812ca9c9c57a0e2.png\" class=\"latex\" alt=\"$1)x^2+xy+y^2+2x-2y=0$\" style=\"vertical-align: -5px\" width=\"233\" height=\"21\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/1\/6\/3\/163561fd6261567ec4e8aa27119c97abfb9648b1.png\" class=\"latex\" alt=\"$2)x^2-4xy+4y^2+5\\sqrt5{y}+1=0$\" style=\"vertical-align: -5px\" width=\"265\" height=\"24\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/e\/0\/8\/e08f6b08e8d9ff74adc2cd44db7c86c7a5085441.png\" class=\"latex\" alt=\"$3)x^2+4xy+y^2-4x+8y+\\frac{2}{3}=0$\" style=\"vertical-align: -13px\" width=\"278\" height=\"38\" >","post_canonical":"Reduce each equation,$1$ through $3$,to standard form. Then find the coordinates of the center ,the vertices , and the foci.\n$1)x^2+xy+y^2+2x-2y=0$\n$2)x^2-4xy+4y^2+5\\sqrt5{y}+1=0$\n$3)x^2+4xy+y^2-4x+8y+\\frac{2}{3}=0$","username":"yt12","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1649748245,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_142747.png","num_posts":678,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2822357,"comment_count":3,"num_deleted":0,"topic_title":"Conic (Find equation in standard form)","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":24942736,"first_poster_id":142747,"first_post_time":1649748245,"first_poster_name":"yt12","last_post_time":1653064808,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_142747.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_92334.jpg","last_post_id":25249877,"last_poster_id":92334,"last_poster_name":"vanstraelen","last_update_time":1653071337,"category_id":7,"is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"tags":[{"tag_id":29605,"tag_text":"conic","is_visible":true},{"tag_id":251,"tag_text":"analytic geometry","is_visible":true},{"tag_id":116,"tag_text":"calculus","is_visible":true},{"tag_id":228,"tag_text":"ellipse","is_visible":true},{"tag_id":229,"tag_text":"hyperbola","is_visible":true},{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":169,"tag_text":"algebra","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Reduce each equation<span style=\"white-space:pre;\">,<img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" ><\/span> through <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/d\/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" >,<\/span>to standard form. Then find the coordinates of the center ,the vertices , and the foci.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/3\/4\/a\/34a1b2b56eef892795b093bab812ca9c9c57a0e2.png\" class=\"latex\" alt=\"$1)x^2+xy+y^2+2x-2y=0$\" style=\"vertical-align: -5px\" width=\"233\" height=\"21\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/1\/6\/3\/163561fd6261567ec4e8aa27119c97abfb9648b1.png\" class=\"latex\" alt=\"$2)x^2-4xy+4y^2+5\\sqrt5{y}+1=0$\" style=\"vertical-align: -5px\" width=\"265\" height=\"24\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/e\/0\/8\/e08f6b08e8d9ff74adc2cd44db7c86c7a5085441.png\" class=\"latex\" alt=\"$3)x^2+4xy+y^2-4x+8y+\\frac{2}{3}=0$\" style=\"vertical-align: -13px\" width=\"278\" height=\"38\" >","category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","num_reports":0,"poll_id":0,"source":"","category_num_users":5,"category_num_topics":102698,"category_num_posts":402685,"num_views":126,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"1652146":{"num_posts":3,"posts_data":[{"post_id":10453587,"topic_id":1652146,"poster_id":281882,"post_rendered":"If <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/2\/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > is a graph of order <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > such that for all distinct nonadjacent vertices <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/d\/9ad99798ec4c38e165cf517cb9e02b1c9e824103.png\" class=\"latex\" alt=\"$u$\" width=\"10\" height=\"8\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/f\/a9f23bf124b6b2b2a993eb313c72e678664ac74a.png\" class=\"latex\" alt=\"$v$\" width=\"9\" height=\"8\" >,<\/span><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/1\/3c186807ba7ca94b416da3e53fef6109ceca0712.png\" class=\"latexcenter\" alt=\"$$d_{G}(u) + d_{G}(v) \\ge n + 1,$$\" width=\"181\" height=\"18\" >then <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/2\/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > is hamiltonian-connected(i.e. any two nonadjacent vertices are connected with hamiltonian path).","post_canonical":"If $G$ is a graph of order $n$ such that for all distinct nonadjacent vertices $u$ and $v$,\n$$d_{G}(u) + d_{G}(v) \\ge n + 1,$$\nthen $G$ is hamiltonian-connected(i.e. any two nonadjacent vertices are connected with hamiltonian path).\n ","username":"mard","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1527868823,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_281882.png","num_posts":4,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":1652146,"comment_count":3,"num_deleted":0,"topic_title":"Graph Theory","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":10453587,"first_poster_id":281882,"first_post_time":1527868823,"first_poster_name":"mard","last_post_time":1653060801,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_281882.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_270065.png?t=1647592393","last_post_id":25249454,"last_poster_id":270065,"last_poster_name":"CeuAzul","last_update_time":1653060801,"category_id":7,"is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"tags":[{"tag_id":240,"tag_text":"graph theory","is_visible":true},{"tag_id":442,"tag_text":"college contests","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"If <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/2\/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > is a graph of order <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > such that for all distinct nonadjacent vertices <img src=\"\/\/latex.artofproblemsolving.com\/9\/a\/d\/9ad99798ec4c38e165cf517cb9e02b1c9e824103.png\" class=\"latex\" alt=\"$u$\" width=\"10\" height=\"8\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/9\/f\/a9f23bf124b6b2b2a993eb313c72e678664ac74a.png\" class=\"latex\" alt=\"$v$\" width=\"9\" height=\"8\" >,<\/span><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/3\/c\/1\/3c186807ba7ca94b416da3e53fef6109ceca0712.png\" class=\"latexcenter\" alt=\"$$d_{G}(u) + d_{G}(v) \\ge n + 1,$$\" width=\"181\" height=\"18\" >then <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/2\/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > is hamiltonian-connected(i.e. any two nonadjacent vertices are connected with hamiltonian path).<br>\n","category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","num_reports":0,"poll_id":0,"source":"","category_num_users":5,"category_num_topics":102698,"category_num_posts":402685,"num_views":214,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[10]={"2848713":{"num_posts":2,"posts_data":[{"post_id":25249521,"topic_id":2848713,"poster_id":221021,"post_rendered":"Have the "Keep Learning" problems been discontinued? There hasn't been anything new there for a few months.","post_canonical":"Have the \"Keep Learning\" problems been discontinued? There hasn't been anything new there for a few months.","username":"sdpandit","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":1,"nothanks_received":0,"thankers":"ImSh95","deleted":false,"post_number":1,"post_time":1653061362,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_221021.png","num_posts":2295,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848713,"comment_count":2,"num_deleted":0,"topic_title":"Keep Learning","target_url":"","target_text":"","topic_type":"forum","state":"closed","first_post_id":25249521,"first_poster_id":221021,"first_post_time":1653061362,"first_poster_name":"sdpandit","last_post_time":1653062542,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_221021.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_219683.png?t=1607040961","last_post_id":25249632,"last_poster_id":219683,"last_poster_name":"devenware","last_update_time":1653062542,"category_id":10,"is_public":true,"roles":{"473145":"mod","500529":"mod","481507":"mod","470423":"mod"},"tags":[{"tag_id":758462,"tag_text":"keeplearning","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Have the "Keep Learning" problems been discontinued? There hasn't been anything new there for a few months.","category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","num_reports":0,"poll_id":0,"source":"","category_num_users":4,"category_num_topics":16579,"category_num_posts":138067,"num_views":43,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848712":{"num_posts":2,"posts_data":[{"post_id":25249489,"topic_id":2848712,"poster_id":883469,"post_rendered":"Hi all,<br>\nI'm trying to edit an AoPS wiki page and I'm logged in, yet I can't edit it because AoPS thinks I'm not logged into the wiki for some stupid reason.<br>\nPlease help!<br>\nThanks!","post_canonical":"Hi all,\nI'm trying to edit an AoPS wiki page and I'm logged in, yet I can't edit it because AoPS thinks I'm not logged into the wiki for some stupid reason.\nPlease help!\nThanks!","username":"hastapasta","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1653061152,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_883469.jpeg?t=1641503781","num_posts":68,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848712,"comment_count":2,"num_deleted":1,"topic_title":"Why can't I edit the wiki when I'm logged in?","target_url":"","target_text":"","topic_type":"forum","state":"closed","first_post_id":25249489,"first_poster_id":883469,"first_post_time":1653061152,"first_poster_name":"hastapasta","last_post_time":1653061674,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_883469.jpeg?t=1641503781","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_944981.png?t=1649452110","last_post_id":25249549,"last_poster_id":944981,"last_poster_name":"jlacosta","last_update_time":1653061685,"category_id":10,"is_public":true,"roles":{"473145":"mod","500529":"mod","481507":"mod","470423":"mod"},"tags":[{"tag_id":45137,"tag_text":"Bug","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false},{"tag_id":30967,"tag_text":"help","is_visible":true},{"tag_id":104956,"tag_text":"technical difficulties","is_visible":true}],"can_have_source":false,"locked":true,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Hi all,<br>\nI'm trying to edit an AoPS wiki page and I'm logged in, yet I can't edit it because AoPS thinks I'm not logged into the wiki for some stupid reason.<br>\nPlease help!<br>\nThanks!","category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","num_reports":0,"poll_id":0,"source":"","category_num_users":4,"category_num_topics":16579,"category_num_posts":138067,"num_views":54,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2848334":{"num_posts":4,"posts_data":[{"post_id":25245172,"topic_id":2848334,"poster_id":593425,"post_rendered":"Hello! I know how to do this: <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Click to reveal hidden text<\/span><div class=\"cmty-hide-content\" style=\"display:none\">insert whatever here<\/div>, but I was wondering how to change the 'click to reveal hidden text' to a different message. Can you please help? Thank you!","post_canonical":"Hello! I know how to do this: [hide]insert whatever here[\/hide], but I was wondering how to change the 'click to reveal hidden text' to a different message. Can you please help? Thank you!","username":"hamster9","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1652999388,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_593425.jpg?t=1648152953","num_posts":339,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2848334,"comment_count":4,"num_deleted":5,"topic_title":"How to label hide tags","target_url":"","target_text":"","topic_type":"forum","state":"closed","first_post_id":25245172,"first_poster_id":593425,"first_post_time":1652999388,"first_poster_name":"hamster9","last_post_time":1652999761,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_593425.jpg?t=1648152953","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_501245.png?t=1641590124","last_post_id":25245214,"last_poster_id":501245,"last_poster_name":"aayr","last_update_time":1653001694,"category_id":10,"is_public":true,"roles":{"473145":"mod","500529":"mod","481507":"mod","470423":"mod"},"tags":[{"tag_id":5,"tag_text":"\/closed","is_visible":false},{"tag_id":95,"tag_text":"Support","is_visible":true},{"tag_id":30808,"tag_text":"Hide Tags","is_visible":true}],"can_have_source":false,"locked":true,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Hello! I know how to do this: <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">Click to reveal hidden text<\/span><div class=\"cmty-hide-content\" style=\"display:none\">insert whatever here<\/div>, but I was wondering how to change the 'click to reveal hidden text' to a different message. Can you please help? Thank you!","category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","num_reports":0,"poll_id":0,"source":"","category_num_users":4,"category_num_topics":16579,"category_num_posts":138067,"num_views":197,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2847864":{"num_posts":4,"posts_data":[{"post_id":25239776,"topic_id":2847864,"poster_id":638097,"post_rendered":"Hi everyone! I was wondering if there was a way for me to see my results on last years AMC 10 and Mathcounts competition?","post_canonical":"Hi everyone! I was wondering if there was a way for me to see my results on last years AMC 10 and Mathcounts competition? ","username":"Justinshao","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1652925144,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_638097.png","num_posts":1,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2847864,"comment_count":4,"num_deleted":0,"topic_title":"AMC 10 and Mathcounts score from past year","target_url":"","target_text":"","topic_type":"forum","state":"closed","first_post_id":25239776,"first_poster_id":638097,"first_post_time":1652925144,"first_poster_name":"Justinshao","last_post_time":1652997903,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_638097.png","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_483700.jpg?t=1564175849","last_post_id":25245043,"last_poster_id":483700,"last_poster_name":"jwelsh","last_update_time":1652997908,"category_id":10,"is_public":true,"roles":{"473145":"mod","500529":"mod","481507":"mod","470423":"mod"},"tags":[{"tag_id":706810,"tag_text":"AMC Scores","is_visible":true},{"tag_id":179,"tag_text":"AMC 10","is_visible":true},{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":true,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Hi everyone! I was wondering if there was a way for me to see my results on last years AMC 10 and Mathcounts competition? ","category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","num_reports":0,"poll_id":0,"source":"","category_num_users":4,"category_num_topics":16579,"category_num_posts":138067,"num_views":287,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}}; |