{"id":13707,"date":"2020-07-22T17:38:30","date_gmt":"2020-07-22T14:38:30","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=13707"},"modified":"2020-09-16T14:15:00","modified_gmt":"2020-09-16T11:15:00","slug":"problem-of-the-month-july-august-2020","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-july-august-2020\/","title":{"rendered":"Problem of the Month \u2013 July, August 2020"},"content":{"rendered":"<p>Let <em>A<sub>1<\/sub>A<sub>2<\/sub>A<sub>3<\/sub>A<sub>4<\/sub><\/em> be a circumscribed quadrilateral with the perimeter <em>p<\/em><sub>1<\/sub>\u00a0and with the sum of<br \/>\nits diagonals <em>k<\/em><sub>1<\/sub> and let <em>B<sub>1<\/sub>B<sub>2<\/sub>B<sub>3<\/sub>B<sub>4<\/sub><\/em> be a circumscribed quadrilateral with the perimeter <em>p<\/em><sub>2<\/sub><br \/>\nand with the sum of its diagonals <em>k<\/em><sub>2<\/sub>. Given<\/p>\n<p style=\"text-align: center;\"><em>p<\/em><sub>1<\/sub><sup>2\u00a0<\/sup>+\u00a0<em>p<\/em><sub>2<\/sub><sup>2\u00a0<\/sup>= (k<sub>1<\/sub>\u00a0+ k<sub>2<\/sub>)<sup>2\u00a0<\/sup><\/p>\n<p>prove that <em>A<sub>1<\/sub>A<sub>2<\/sub>A<sub>3<\/sub>A<sub>4<\/sub><\/em> and <em>B<sub>1<\/sub>B<sub>2<\/sub>B<sub>3<\/sub>B<sub>4<\/sub><\/em> are congruent squares.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Correct Solutions by,<\/strong><\/p>\n<ul>\n<li>Toshihiro Shimizu, Kawasaki, Japan<\/li>\n<li>Haruhisa\u00a0\u00a0 Oda,\u00a0University of Tokyo, Japan<\/li>\n<li>Alper Balc\u0131, METU<\/li>\n<li>Mehmet \u00d6zg\u00fcn Cihangir, METU<\/li>\n<li>Roger Bengtsson, Lund, Sweden<\/li>\n<li>Emirhan Ya\u011fc\u0131o\u011flu,\u00a0\u00d6zel Ankara Matematik ve Fen Bilimleri Anadolu Lisesi<\/li>\n<li>Vedat Deveci, \u0130stanbul<\/li>\n<li>\u0130brahim Berkay Ceylan,\u00a0Giresun Fen Lisesi<\/li>\n<li>Chui Tsz Fung, Hong Kong<\/li>\n<li>Kadyr Annabayev,\u00a0Turkmen State Institute of Arhcitecture and Construction,\u00a0 A\u015fkabat<\/li>\n<li>Ahmed Said Gen\u00e7kol,\u00a0Ankara Atat\u00fcrk Anadolu Lisesi<\/li>\n<li>Mehmet Koca,\u00a0Denizli Koleji<\/li>\n<\/ul>\n<p><strong>Solution:\u00a0<a href=\"http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/2008a.pdf\">http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/2008a.pdf<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let A1A2A3A4 be a circumscribed quadrilateral with the perimeter p1\u00a0and with the sum of its diagonals k1 and let B1B2B3B4 be a circumscribed quadrilateral with the perimeter p2 and with the sum of its diagonals k2. Given p12\u00a0+\u00a0p22\u00a0= (k1\u00a0+ k2)2\u00a0 prove that A1A2A3A4 and B1B2B3B4 are congruent squares. &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-july-august-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-13707","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/13707","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=13707"}],"version-history":[{"count":4,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/13707\/revisions"}],"predecessor-version":[{"id":13890,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/13707\/revisions\/13890"}],"wp:attachment":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/media?parent=13707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}