{"id":206,"date":"2018-11-03T10:50:40","date_gmt":"2018-11-03T15:50:40","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=206"},"modified":"2018-11-09T12:53:07","modified_gmt":"2018-11-09T18:53:07","slug":"problem-8-find-the-largest-product-november-2-2018","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2018\/11\/03\/problem-8-find-the-largest-product-november-2-2018\/","title":{"rendered":"Problem 8 \u2013 Find the Largest Product \u2013 November 2, 2018"},"content":{"rendered":"<p><span style=\"font-size: xx-large\"><b>Find The Largest Product<\/b><\/span><\/p>\n<p><i>This problem is from Zachary M. Franco, Butler University.<\/i><\/p>\n<p>Write 53 as a sum of positive <i><b>real<\/b><\/i> numbers so that their product is as large as possible.<\/p>\n<p>Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found\u00a0<a href=\"http:\/\/blogs.acu.edu\/mu_sigma\/2018\/08\/28\/problem-of-the-week-competition\/\">here<\/a>.<\/p>\n<p><span style=\"font-size: large\">Solution to <b>Find The Largest Product<\/b><\/span><\/p>\n<p><span style=\"font-size: small\"><i>Correct solutions were submitted by: Wyatt Witemeyer.<\/i><\/span><\/p>\n<p>If <i>S = x + y<\/i>, then you can always do better by replacing <i>x<\/i> and <i>y<\/i> with their average. Thus, the answer will consist of <i>n<\/i> pieces all equal to 53\/<i>n<\/i>.<\/p>\n<p>Since the function (53\/<i>x<\/i>)<sup><i>x<\/i><\/sup> has a maximum at approximately <i>x<\/i> = 19.5, we test the nearest discrete cases to determine that the largest product is (53\/19)<sup>19<\/sup> &gt; 291,691,050.<\/p>\n<p>It is interesting to observe that for all integers <i>k<\/i> from 1 to 52, the largest product is (<i>k<\/i>\/<i>n<\/i>)<sup><i>n<\/i><\/sup>, where <i>n<\/i> is such that <i>k\/n<\/i> is closest to <i>e<\/i>. But this fails for 53. For 53, we use <i>n<\/i> where <i>n<\/i> is the closest integer to <i>k\/e<\/i>.<\/p>\n<p>The counterexample list is: 53, 246, 439, 632, 12973, &#8230; (which seems to arise from the continued fraction for <i>e<\/i>). See Z. Franco, <i>Mathematics Magazine<\/i>, Feb 2000, for a proof of the theorem, &#8220;use <i>n<\/i> where <i>n<\/i> is the closest integer to <i>k\/e<\/i>,&#8221; and for the connection to continued fractions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find The Largest Product This problem is from Zachary M. Franco, Butler University. Write 53 as a sum of positive real numbers so that their product is as large as possible. Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found\u00a0here. Solution to Find The Largest Product Correct solutions were submitted by: Wyatt Witemeyer. [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-206","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=206"}],"version-history":[{"count":2,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/206\/revisions"}],"predecessor-version":[{"id":210,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/206\/revisions\/210"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}