{"id":23478,"date":"2024-12-15T14:36:51","date_gmt":"2024-12-15T11:36:51","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=23478"},"modified":"2025-01-31T00:56:15","modified_gmt":"2025-01-30T21:56:15","slug":"problem-of-the-month-december-2024","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-december-2024\/","title":{"rendered":"Problem of the Month \u2013 December 2024"},"content":{"rendered":"<p>Let <em>S<\/em> be a set consisting of 31 positive real numbers. For each non-empty subset <em>A<\/em> \u2282 <em>S<\/em> let<br \/>\n<em>f<\/em>(<em>A<\/em>) be the product of all elements of <em>A<\/em>. We say that a subset <em>A<\/em> \u2282 <em>S<\/em> is rational if <em>f<\/em>(<em>A<\/em>) is<br \/>\na rational number. We say that a subset <em>A<\/em> \u2282 <em>S<\/em> is irrational if <em>f<\/em>(<em>A<\/em>) is an irrational number.<br \/>\nIs there any set <em>S<\/em> having exactly 2023 rational subsets? Is there any set <em>S<\/em> having exactly<br \/>\n2025 irrational subsets?<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Correct Solutions by,<\/strong><\/p>\n<ul>\n<li>Toshihiro Shimizu Kawasaki, Japan<\/li>\n<li>Magnus Jakobsson Lund, Sweden<\/li>\n<li>Jepbar Asgarov\u00a0A\u015fgabat, Turkmenistan<\/li>\n<li>Alper Balc\u0131\u00a0Ohio State University<\/li>\n<li>Hamza Akdeniz\u00a0Bilkent University<\/li>\n<li>Mehmet Samet \u00d6d\u00fck METU<\/li>\n<li>Arda Karahan\u00a0Trabzon Fen Lisesi<\/li>\n<\/ul>\n<p><strong>Solution:\u00a0<a href=\"http:\/\/www.fen.bilkent.edu.tr\/~cvmath\/Problem\/2412a.pdf\">2412a.pdf (bilkent.edu.tr)<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let S be a set consisting of 31 positive real numbers. For each non-empty subset A \u2282 S let f(A) be the product of all elements of A. We say that a subset A \u2282 S is rational if f(A) is a rational number. We say that a subset A [\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-2024\/\" 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-23478","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/23478","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=23478"}],"version-history":[{"count":3,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/23478\/revisions"}],"predecessor-version":[{"id":23729,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/23478\/revisions\/23729"}],"wp:attachment":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/media?parent=23478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}