{"id":1204,"date":"2019-08-10T08:22:32","date_gmt":"2019-08-10T12:22:32","guid":{"rendered":"http:\/\/josmfs.net\/?p=1204"},"modified":"2019-08-10T08:22:32","modified_gmt":"2019-08-10T12:22:32","slug":"maximum-product","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/08\/10\/maximum-product\/","title":{"rendered":"Maximum Product"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-1202\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/08\/Maximum-Product-fig.jpg\" alt=\"\" width=\"212\" height=\"124\" \/>This 2007 four-star problem from Colin Hughes at <em>Maths Challenge<\/em> is definitely a bit challenging.<\/p>\n<p>\u201c<strong>Problem<\/strong><br \/>\nFor any positive integer, k, let Sk = {x1, x2, &#8230; , xn} be the set of [non-negative] real numbers for which x1 + x2 + &#8230; + xn = k and P = x1 x2 &#8230; xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.<\/p>\n<p>Prove that P is maximised when all the elements of S are equal in value and rational.\u201d<\/p>\n<p>I took a different approach from Maths Challenge, but for me, it did not rely on remembering a somewhat obscure formula. (I don\u2019t remember formulas well at my age\u2014only procedures, processes, or proofs, which is ironic, since at a younger age it was just the opposite.) It is also clear from the Maths Challenge solution that the numbers were assumed to be non-negative.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/08\/Maximum-Product-190803.pdf\">Maximum Product<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This 2007 four-star problem from Colin Hughes at Maths Challenge is definitely a bit challenging. \u201cProblem For any positive integer, k, let Sk = {x1, x2, &#8230; , xn} be the set of [non-negative] real numbers for which x1 + x2 + &#8230; + xn = k and P = x1 x2 &#8230; xn is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[32,151,153],"class_list":["post-1204","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-calculus","tag-colin-hughes","tag-optimization"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1204","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1204"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1204\/revisions"}],"predecessor-version":[{"id":1205,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1204\/revisions\/1205"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1204"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1204"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}