1 line
192 KiB
JavaScript
1 line
192 KiB
JavaScript
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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":0,"poll_id":44851,"source":"","category_num_users":14,"category_num_topics":46945,"category_num_posts":659609,"num_views":6466,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2842228":{"num_posts":67,"posts_data":[{"post_id":25178045,"topic_id":2842228,"poster_id":350053,"post_rendered":"teamwork and friendship is magic","post_canonical":"teamwork and friendship is magic","username":"DottedCaculator","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":44,"nothanks_received":0,"thankers":"rg_ryse, asimov, pearlhaas, math31415926535, Gacac, wxl18, mathking999, ilovepizza2020, megarnie, GeoWhiz4536, tenebrine, player01, mahaler, mathmusician, akpi2, HWenslawski, YaoAOPS, E.C_SilverStangs, rayfish, marinasimp, Bimikel, 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summary","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25178045,"first_poster_id":350053,"first_post_time":1652206302,"first_poster_name":"DottedCaculator","last_post_time":1653073512,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_350053.png?t=1642719368","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_945473.jpg?t=1652224039","last_post_id":25250603,"last_poster_id":945473,"last_poster_name":"Cygnet","last_update_time":1653073512,"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":"teamwork and friendship is magic","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":0,"poll_id":0,"source":"","category_num_users":14,"category_num_topics":46945,"category_num_posts":659609,"num_views":2874,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"2843284":{"num_posts":52,"posts_data":[{"post_id":25190306,"topic_id":2843284,"poster_id":541699,"post_rendered":"Make the largest number you can under the following rules:<br>\n\n<ul class=\"bbcode_list\">\n<li>You must use each of the integers <img src=\"\/\/latex.artofproblemsolving.com\/5\/3\/4\/53458ebc12bfd4c66c5cad57012a1200803297e1.png\" class=\"latex\" alt=\"$0-9$\" width=\"40\" height=\"12\" > exactly once.<\/li>\n<li>You are allowed to use any of the four operations <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/7\/c\/77cf26be132ef93923e082ee4153b2cb0ef44a50.png\" class=\"latex\" alt=\"$+$\" style=\"vertical-align: -1px\" width=\"13\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/d\/add22d9bda21e86390bc74fa4dde17730a442da7.png\" class=\"latex\" alt=\"$-$\" style=\"vertical-align: -2px\" width=\"16\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/c\/8\/1c81486a525ba08192aab8546dbe9a32bdbe0c07.png\" class=\"latex\" alt=\"$\\div$\" style=\"vertical-align: -2px\" width=\"16\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/a\/c\/bac4dbe1c696d11e8dc43dd7f613199b2120daa1.png\" class=\"latex\" alt=\"$\\times$\" width=\"11\" height=\"9\" >,<\/span> and parentheses<\/li>\n<li>You are allowed to use exponentiation<\/li>\n<li>You are allowed to use <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">concatenation<\/span><div class=\"cmty-hide-content\" style=\"display:none\">denote concatenation using <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/9\/5\/b95f37a176047e59b4af0f347d1e9d07e3a61fa9.png\" class=\"latex\" alt=\"$||$\" style=\"vertical-align: -4px\" width=\"8\" height=\"18\" >,<\/span> so <img src=\"\/\/latex.artofproblemsolving.com\/a\/b\/d\/abdac5abb5c13aab8217b66ad84d699e53acd54e.png\" class=\"latex\" alt=\"$4 || 7=47$\" style=\"vertical-align: -5px\" width=\"73\" height=\"20\" ><\/div><\/li>\n<li>and you are allowed to use logarithms, although the base must be written except when using the natural log<\/li>\n<\/ul>\n<br>\nThen, try doing it if you are allowed to use <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">tetration<\/span><div class=\"cmty-hide-content\" style=\"display:none\">for tetration, denote tetration as <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/b\/7\/3b797951a62e0cbaea42fd4545a5f396565b3907.png\" class=\"latex\" alt=\"$b^{b^{b^{\\cdot^{\\cdot^{\\cdot^ b}}}}}=^ab$\" style=\"vertical-align: -1px\" width=\"82\" height=\"34\" >,<\/span> where there are <img src=\"\/\/latex.artofproblemsolving.com\/c\/7\/d\/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" > <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/1\/3\/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" style=\"vertical-align: -1px\" width=\"10\" height=\"14\" >'<\/span>s in the exponentiation tower. for example, <img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/9\/37970bb948c4ca0aadd1d4f1e9443d646b5d9dfb.png\" class=\"latex\" alt=\"$^37=7^{7^7}$\" style=\"vertical-align: -1px\" width=\"66\" height=\"20\" ><\/div>","post_canonical":"Make the largest number you can under the following rules:\n\n[list]\n[*] You must use each of the integers $0-9$ exactly once.\n[*] You are allowed to use any of the four operations $+$, $-$, $\\div$, $\\times$, and parentheses\n[*]You are allowed to use exponentiation\n[*] You are allowed to use [hide=concatenation] denote concatenation using $||$, so $4 || 7=47$[\/hide]\n[*] and you are allowed to use logarithms, although the base must be written except when using the natural log\n[\/list]\n\nThen, try doing it if you are allowed to use [hide=tetration]for tetration, denote tetration as $b^{b^{b^{\\cdot^{\\cdot^{\\cdot^ b}}}}}=^ab$, where there are $a$ $b$'s in the exponentiation tower. for example, $^37=7^{7^7}$[\/hide]","username":"eagles2018","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":7,"nothanks_received":0,"thankers":"violin21, tennisrules, HWenslawski, ImSh95, asimov, son7, TsunamiStorm08","deleted":false,"post_number":1,"post_time":1652322082,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_541699.png?t=1594928893","num_posts":2292,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":2843284,"comment_count":52,"num_deleted":3,"topic_title":"Challenge: The biggest number","target_url":"","target_text":"","topic_type":"forum","state":"none","first_post_id":25190306,"first_poster_id":541699,"first_post_time":1652322082,"first_poster_name":"eagles2018","last_post_time":1653070737,"first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_541699.png?t=1594928893","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_402075.jpg?t=1631296469","last_post_id":25250384,"last_poster_id":402075,"last_poster_name":"fruitmonster97","last_update_time":1653070737,"category_id":3,"is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"tags":[{"tag_id":31454,"tag_text":"maximum","is_visible":true},{"tag_id":177,"tag_text":"number theory","is_visible":true},{"tag_id":30508,"tag_text":"challenge","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Make the largest number you can under the following rules:<br>\n<br>\n[list]<br>\n[*] You must use each of the integers <img src=\"\/\/latex.artofproblemsolving.com\/5\/3\/4\/53458ebc12bfd4c66c5cad57012a1200803297e1.png\" class=\"latex\" alt=\"$0-9$\" width=\"40\" height=\"12\" > exactly once.<br>\n[*] You are allowed to use any of the four operations <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/7\/c\/77cf26be132ef93923e082ee4153b2cb0ef44a50.png\" class=\"latex\" alt=\"$+$\" style=\"vertical-align: -1px\" width=\"13\" height=\"12\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/d\/d\/add22d9bda21e86390bc74fa4dde17730a442da7.png\" class=\"latex\" alt=\"$-$\" style=\"vertical-align: -2px\" width=\"16\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/c\/8\/1c81486a525ba08192aab8546dbe9a32bdbe0c07.png\" class=\"latex\" alt=\"$\\div$\" style=\"vertical-align: -2px\" width=\"16\" height=\"14\" >,<\/span> <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/a\/c\/bac4dbe1c696d11e8dc43dd7f613199b2120daa1.png\" class=\"latex\" alt=\"$\\times$\" width=\"11\" height=\"9\" >,<\/span> and parentheses<br>\n[*]You are allowed to use exponentiation<br>\n[*] You are allowed to use <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">concatenation<\/span><div class=\"cmty-hide-content\" style=\"display:none\">denote concatenation using <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/9\/5\/b95f37a176047e59b4af0f347d1e9d07e3a61fa9.png\" class=\"latex\" alt=\"$||$\" style=\"vertical-align: -4px\" width=\"8\" height=\"18\" >,<\/span> so <img src=\"\/\/latex.artofproblemsolving.com\/a\/b\/d\/abdac5abb5c13aab8217b66ad84d699e53acd54e.png\" class=\"latex\" alt=\"$4 || 7=47$\" style=\"vertical-align: -5px\" width=\"73\" height=\"20\" ><\/div><br>\n[*] and you are allowed to use logarithms, although the base must be written except when using the natural log<br>\n[\/list]<br>\n<br>\nThen, try doing it if you are allowed to use <span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">tetration<\/span><div class=\"cmty-hide-content\" style=\"display:none\">for tetration, denote tetration as <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/b\/7\/3b797951a62e0cbaea42fd4545a5f396565b3907.png\" class=\"latex\" alt=\"$b^{b^{b^{\\cdot^{\\cdot^{\\cdot^ b}}}}}=^ab$\" style=\"vertical-align: -1px\" width=\"82\" height=\"34\" >,<\/span> where there are <img src=\"\/\/latex.artofproblemsolving.com\/c\/7\/d\/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" > <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/1\/3\/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" style=\"vertical-align: -1px\" width=\"10\" height=\"14\" >'<\/span>s in the exponentiation tower. for example, <img src=\"\/\/latex.artofproblemsolving.com\/3\/7\/9\/37970bb948c4ca0aadd1d4f1e9443d646b5d9dfb.png\" class=\"latex\" alt=\"$^37=7^{7^7}$\" style=\"vertical-align: -1px\" width=\"66\" height=\"20\" ><\/div>","category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","num_reports":0,"poll_id":0,"source":"","category_num_users":14,"category_num_topics":46945,"category_num_posts":659609,"num_views":2153,"cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[4]={"2848786":{"num_posts":1,"posts_data":[{"post_id":25250572,"topic_id":2848786,"poster_id":167643,"post_rendered":"<b>VII.<\/b> <u>Rounds 1-4<\/u><br>\n<br>\n<br>\n<b>1.1 \/ S1.1<\/b> What is the parity of the number of ways to cut a rectangle into rectangles?<br>\n<br>\n<br>\n<b>1.2 \/ S1.2<\/b> In an isosceles triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$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? 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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}}; |