{"id":11539,"date":"2019-06-10T10:08:44","date_gmt":"2019-06-10T07:08:44","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=11539"},"modified":"2019-07-10T09:16:08","modified_gmt":"2019-07-10T06:16:08","slug":"problem-of-the-month-june-2019","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-june-2019\/","title":{"rendered":"Problem of the Month \u2013 June 2019"},"content":{"rendered":"<p>Let <em>P(x)<\/em> be a non-constant polynomial with real coefficients such that all of its roots are<br \/>\nreal numbers. Suppose that there exists a polynomial <em>Q(x)<\/em> with real coefficients such<br \/>\nthat<\/p>\n<p style=\"padding-left: 150px;\">(<em>P(x)<\/em>)<sup>2<\/sup> = <em>P<\/em>(<em>Q(x)<\/em>)<\/p>\n<p>for all real numbers<em> x<\/em>. Determine the maximal possible number of distinct roots of <em>P(x)<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Correct Solutions by,<\/strong><\/p>\n<ul>\n<li>Toshihiro Shimizu Kawasaki, Japan<\/li>\n<li>Steffen Weber\u00a0Isman\u0131ng, Germany<\/li>\n<li>Max Nilsson\u00a0Lund, Sweden<\/li>\n<li>Bora Ege Duygun\u00a0Bilkent University<\/li>\n<li>Vedat Deveci\u00a0\u0130stanbul<\/li>\n<li>Mehmet Yeni\u00a0Tarsus, Mersin<\/li>\n<li>Halil \u00d6zkan\u00a0Denizli \u00d6zel Denizli Koleji,\u00a0 Denizli<\/li>\n<li>Serdar Hojayev\u00a0Dashoguz, Turkmenistan<\/li>\n<li>Halil Alperen G\u00f6zeten\u00a0Denizli \u00d6zel Denizli Koleji,\u00a0 Denizli<\/li>\n<li>Demir Eken\u00a0Bilkent University<\/li>\n<li>Bilge K\u00f6ksal\u00a0Bilkent University<\/li>\n<li>Ozan Kaymak\u00a0Denizli \u00d6zel Denizli Koleji,\u00a0 Denizli<\/li>\n<li>Roger Bengtsson\u00a0Lund, Sweden<\/li>\n<li>Hakan Karaku\u015f\u00a0Antalya Yusuf Ziya \u00d6ner Fen Lisesi<\/li>\n<li>\u00d6mer Topalo\u011flu\u00a0\u00d6zel \u0130zmir Fen Lisesi,\u00a0 \u0130zmir<\/li>\n<li>Mustafa Emir \u00c7elebi\u00a0\u0130stanbul Lisesi<\/li>\n<li>\u0130brahim Suat Evren\u00a0Denizli \u00d6zel Denizli Koleji,\u00a0 Denizli<\/li>\n<li>Enes \u00d6zdemir\u00a0Buca \u0130nci \u00d6zer T\u0131rnakl\u0131 Fen Lisesi, \u0130zmir<\/li>\n<li>John Wright\u00a0Brooklyn, New York, USA<\/li>\n<li>J\u00fcrgen Weith\u00a0University of Applied Sciences, Schweinfurt, Germany<\/li>\n<\/ul>\n<p><strong>Solution:\u00a0<\/strong><a href=\"http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/1906a.pdf\">http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/1906a.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let P(x) be a non-constant polynomial with real coefficients such that all of its roots are real numbers. Suppose that there exists a polynomial Q(x) with real coefficients such that (P(x))2 = P(Q(x)) for all real numbers x. Determine the maximal possible number of distinct roots of P(x). &nbsp; Correct [\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-june-2019\/\" 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-11539","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/11539","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=11539"}],"version-history":[{"count":3,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/11539\/revisions"}],"predecessor-version":[{"id":11664,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/11539\/revisions\/11664"}],"wp:attachment":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/media?parent=11539"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}