{"id":931,"date":"2019-04-16T14:30:43","date_gmt":"2019-04-16T18:30:43","guid":{"rendered":"http:\/\/josmfs.net\/?p=931"},"modified":"2024-04-10T19:02:01","modified_gmt":"2024-04-10T23:02:01","slug":"challenging-sum","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/04\/16\/challenging-sum\/","title":{"rendered":"Challenging Sum"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-929\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Challenging-Sum-fig.jpg\" alt=\"\" width=\"186\" height=\"38\" \/>Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:<\/p>\n<p style=\"text-align: left;\">\u201cFour positive integers <em>a, b, c<\/em>, and <em>d<\/em> are such that<\/p>\n<p style=\"text-align: center;\"><em>abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d<\/em> = 2009.<\/p>\n<p style=\"text-align: left;\">What is the value of <em>a + b + c + d<\/em>?<\/p>\n<p style=\"text-align: center;\">A 73<span style=\"color: #ffffff;\">_________<\/span>B 75<span style=\"color: #ffffff;\">_________<\/span>C 77<span style=\"color: #ffffff;\">_________<\/span>D 79<span style=\"color: #ffffff;\">_________<\/span>E 81\u201d<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Challenging-Sum-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Challenging-Sum-190416.pdf\">Challenging Sum<\/a> for a solution.<\/p>\n<p><strong>(Update 4\/17\/2019)<\/strong><!--more-->\u00a0I forgot to mention in my write-up that the UKMT solution only included the last few steps and did not provide any motivation for <em>why<\/em> one would think of it.<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009: \u201cFour positive integers a, b, c, and d are such that abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[133,83],"class_list":["post-931","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-algebra","tag-ukmt-challenge"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/931","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=931"}],"version-history":[{"count":7,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/931\/revisions"}],"predecessor-version":[{"id":3851,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/931\/revisions\/3851"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=931"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=931"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}