{"id":14444,"date":"2020-12-09T10:42:32","date_gmt":"2020-12-09T07:42:32","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=14444"},"modified":"2021-01-11T11:32:27","modified_gmt":"2021-01-11T08:32:27","slug":"problem-of-the-month-december-2020","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-december-2020\/","title":{"rendered":"Problem of the Month \u2013 December 2020"},"content":{"rendered":"<p>Let <em>N<\/em> be the total number of bijective functions<br \/>\n<em>f<\/em> : {1, 2, . . . , 2020} \u2192 {1, 2, . . . , 2020}<br \/>\nsatisfying <em>f\u00a0<\/em>(<em>f\u00a0<\/em>(<em>f\u00a0<\/em>(<em>k<\/em>))) = <em>k<\/em> for all <em>k<\/em> = 1, 2, . . . , 2020. Show that N is divisible by 3<sup>336\u00a0<\/sup>.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Correct Solutions by,<\/strong><\/p>\n<ul>\n<li>Toshihiro Shimizu Kawasaki, Japan<\/li>\n<li>Chui Tsz Fung,\u00a0Hong Kong<\/li>\n<li>Steffen Weber,\u00a0Ismaning, Germany<\/li>\n<li>Albert Stadler,\u00a0Herrliberg, Switzerland<\/li>\n<li>\u0130brahim Berkay Ceylan,\u00a0Giresun Fen Lisesi<\/li>\n<li>Jepbar Asgarov,\u00a0Hitrowka, Turkmenistan<\/li>\n<li>Tan \u00d6zalp,\u00a0Orta Do\u011fu Teknik \u00dcniversitesi, Ankara<\/li>\n<li>Magnus Jakobsson,\u00a0Lund, Sweden<\/li>\n<li>Muhammed \u0130kbal Ulvi,\u00a0Orta Do\u011fu Teknik \u00dcniversitesi, Ankara<\/li>\n<li>Kadyr Annabayev,\u00a0Turkmen State Institute of Architecture and Construction, Turkmenistan<\/li>\n<li>Roger Bengtsson,\u00a0Lund, Sweden<\/li>\n<li>C\u00fcneyd \u00d6zt\u00fcrk,\u00a0Bilkent \u00dcniversitesi<\/li>\n<li>Omar Sonebi,\u00a0Lycee technique Settat, Casablanca,\u00a0Morocco<\/li>\n<li>Enes Kocabey,\u00a0Google, Mountain View, USA<\/li>\n<li>Kipp Johnson,\u00a0Oregon, USA<\/li>\n<\/ul>\n<p><strong>Solution: <\/strong><a href=\"http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/2012a.pdf\">http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Soru\/2011c.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let N be the total number of bijective functions f : {1, 2, . . . , 2020} \u2192 {1, 2, . . . , 2020} satisfying f\u00a0(f\u00a0(f\u00a0(k))) = k for all k = 1, 2, . . . , 2020. Show that N is divisible by 3336\u00a0. &nbsp; &nbsp; [\u2026] <\/p>\n<div class=\"clear\"><\/div>\n<p><a class=\"more_link clearfix\" href=\"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-december-2020\/\" rel=\"nofollow\">Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-14444","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/14444","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/comments?post=14444"}],"version-history":[{"count":5,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/14444\/revisions"}],"predecessor-version":[{"id":14583,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/14444\/revisions\/14583"}],"wp:attachment":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/media?parent=14444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}