-/Download/​​​ A Mathematical Curiosity_files/community_data.js

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,post_data : {"post_id":3459747,"post_type":"forum","topic_id":585057,"poster_id":146965,"post_rendered":"Determine all real constants <img src=\"\/\/latex.artofproblemsolving.com\/e\/0\/d\/e0d2bf360290fd61d1c1557e763f2622363b3d35.png\" class=\"latex\" alt=\"$t$\"  width=\"6\" height=\"11\" > such that whenever <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/c\/7\/d\/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\"  width=\"9\" height=\"8\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/8\/1\/3\/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\"  width=\"8\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/3\/3\/7\/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png\" class=\"latex\" alt=\"$c$\"  width=\"8\" height=\"8\" > are the lengths of sides of a triangle, then so are <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/2\/d\/f2d8108bdff76b490b211a92fff96443dad92a15.png\" class=\"latex\" alt=\"$a^2+bct$\" style=\"vertical-align: -1px\" width=\"61\" height=\"16\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/9\/1\/e9133c6dd8ed572e9694fb70c27e825b78d98d72.png\" class=\"latex\" alt=\"$b^2+cat$\" style=\"vertical-align: -1px\" width=\"61\" height=\"16\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/5\/8\/b583284bde2729e1295d36e76433e81a76a8bf09.png\" class=\"latex\" alt=\"$c^2+abt$\" style=\"vertical-align: -1px\" width=\"61\" height=\"16\" >.<\/span>","post_canonical":"Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.","username":"shivangjindal","post_number":1,"post_format":"bbcode","category_name":"High School Olympiads","category_id":6,"poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_146965.png"}});
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,post_data : {"post_id":3459750,"post_type":"forum","topic_id":585058,"poster_id":146965,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\"  width=\"15\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\"  width=\"14\" height=\"12\" > be points in the interiors of sides <img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\"  width=\"27\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\"  width=\"27\" height=\"13\" >,<\/span> respectively, of a triangle <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\"  width=\"42\" height=\"13\" >,<\/span> such that <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/9\/5\/5954c8ce56621e52bd7465bd85aba65b872ddd29.png\" class=\"latex\" alt=\"$DB = BC = CE$\"  width=\"136\" height=\"12\" >.<\/span> Let the lines <img src=\"\/\/latex.artofproblemsolving.com\/0\/9\/4\/0948661e822f2a953c43a57ac9e40b2734476de4.png\" class=\"latex\" alt=\"$CD$\"  width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/a\/7\/b\/a7baf95ecafc2d7791788ec79aeeae390d60b119.png\" class=\"latex\" alt=\"$BE$\"  width=\"28\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/0\/5\/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\"  width=\"14\" height=\"12\" >.<\/span> Prove that the incentre <img src=\"\/\/latex.artofproblemsolving.com\/0\/2\/7\/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\"  width=\"9\" height=\"12\" > of triangle <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\"  width=\"42\" height=\"13\" >,<\/span> the orthocentre <img src=\"\/\/latex.artofproblemsolving.com\/b\/1\/9\/b1902d279ba37d60bdce4e0e987b7cd19d48974e.png\" class=\"latex\" alt=\"$H$\"  width=\"16\" height=\"12\" > of triangle <img src=\"\/\/latex.artofproblemsolving.com\/2\/e\/9\/2e9e8b2c23e0545ef6de81f0ea578173e9ab7a64.png\" class=\"latex\" alt=\"$DEF$\"  width=\"44\" height=\"12\" > and the midpoint <img src=\"\/\/latex.artofproblemsolving.com\/5\/d\/1\/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\"  width=\"19\" height=\"12\" > of the arc <img src=\"\/\/latex.artofproblemsolving.com\/2\/e\/d\/2edcdc0d73ef64ec441f60a3a8dccdbd10aa61a5.png\" class=\"latex\" alt=\"$BAC$\"  width=\"42\" height=\"13\" > of the circumcircle of triangle <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\"  width=\"42\" height=\"13\" > are collinear.","post_canonical":"Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.","username":"shivangjindal","post_number":1,"post_format":"bbcode","category_name":"High School Olympiads","category_id":6,"poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_146965.png"}});
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,post_data : {"post_id":3459754,"post_type":"forum","topic_id":585060,"poster_id":146965,"post_rendered":"We denote the number of positive divisors of a positive integer <img src=\"\/\/latex.artofproblemsolving.com\/f\/5\/0\/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\"  width=\"15\" height=\"8\" > by <img src=\"\/\/latex.artofproblemsolving.com\/e\/3\/1\/e31583b08d7a4d2cfef6b1686c943f75bb920c49.png\" class=\"latex\" alt=\"$d(m)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > and the number of distinct prime divisors of <img src=\"\/\/latex.artofproblemsolving.com\/f\/5\/0\/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\"  width=\"15\" height=\"8\" > by <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/1\/9\/919dc0c0fa599acc4c13d5e8a75e19b33182f6f5.png\" class=\"latex\" alt=\"$\\omega(m)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" >.<\/span> Let <img src=\"\/\/latex.artofproblemsolving.com\/8\/c\/3\/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\"  width=\"9\" height=\"12\" > be a positive integer. Prove that there exist infinitely many positive integers <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\"  width=\"10\" height=\"8\" > such that <img src=\"\/\/latex.artofproblemsolving.com\/2\/c\/9\/2c9efe77e963a45b695222612888db8539c25243.png\" class=\"latex\" alt=\"$\\omega(n) = k$\" style=\"vertical-align: -4px\" width=\"71\" height=\"18\" > and <img src=\"\/\/latex.artofproblemsolving.com\/b\/0\/5\/b05725a6508fe8df761c6e91d2735b697ebd87d9.png\" class=\"latex\" alt=\"$d(n)$\" style=\"vertical-align: -4px\" width=\"33\" height=\"18\" > does not divide <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/b\/17b82e2b28496651fd8a395028161093503285a7.png\" class=\"latex\" alt=\"$d(a^2+b^2)$\" style=\"vertical-align: -4px\" width=\"76\" height=\"19\" > for any positive integers <img src=\"\/\/latex.artofproblemsolving.com\/3\/8\/3\/3834f0cef2a1f4d8524aa5bc8e3faca298550862.png\" class=\"latex\" alt=\"$a, b$\" style=\"vertical-align: -3px\" width=\"25\" height=\"16\" > satisfying <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/d\/d6d145c0bf2d95f16f578fbd3045d9883173bcf0.png\" class=\"latex\" alt=\"$a + b = n$\" style=\"vertical-align: -1px\" width=\"74\" height=\"14\" >.<\/span>","post_canonical":"We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.","username":"shivangjindal","post_number":1,"post_format":"bbcode","category_name":"High School Olympiads","category_id":6,"poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_146965.png"}});
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