{"id":239,"date":"2019-02-22T14:14:08","date_gmt":"2019-02-22T20:14:08","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=239"},"modified":"2019-03-01T15:43:48","modified_gmt":"2019-03-01T21:43:48","slug":"problem-13-february-22nd-not-that-far-away","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2019\/02\/22\/problem-13-february-22nd-not-that-far-away\/","title":{"rendered":"Problem 13 &#8211; February 22nd &#8211; Not That Far Away!"},"content":{"rendered":"<p><b>Problem 13 \u2013 <\/b>February 22, 2019<\/p>\n<p><span style=\"font-size: xx-large\"><b>Some Are Not That Far Away<\/b><\/span><\/p>\n<p>Consider any five points <i>P<\/i><sub>1<\/sub>, <i>P<\/i><sub>2<\/sub>, <i>P<\/i><sub>3<\/sub>, <i>P<\/i><sub>4<\/sub>, and <i>P<\/i><sub>5<\/sub> in the interior of a square of side length 1. Denote by <i>d<\/i><sub>ij<\/sub> the distance between points <i>P<\/i><sub>i<\/sub> and <i>P<\/i><sub>j<\/sub> when <i>i<\/i> is not equal to <i>j<\/i>. Prove that at least one of the distances <i>d<\/i><sub>ij<\/sub> is less than <sub><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/math.acu.edu\/green\/pow\/20031\/pow04im1.gif\" width=\"28\" height=\"45\" align=\"bottom\" \/><\/sub>.<\/p>\n<p>Your answer must be submitted as a proof!<\/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>Correct answers were submitted by Dr. Mark Riggs!<\/p>\n<p><span style=\"font-size: large\">Solution to <b>Some Are Not That Far Away<\/b><\/span><\/p>\n<p>The unit square may be divided into four square quadrants, each with side length 1\/2. Each point must be in one of these quadrants, and since there are five points, two must be in the same quadrant.<\/p>\n<p>The farthest apart two points in a square can be is the length of a diagonal, which is achieved when the points are at opposite corners. But two opposite corners of a quadrant must have at least one on the border of the unit square. So for both to be in the interior of the unit square, they must be even closer. Therefore the distance between two of the points is less than the length of a diagonal of a square of side 1\/2, which is <sub><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/math.acu.edu\/green\/pow\/20031\/pow04im1.gif\" width=\"28\" height=\"45\" align=\"bottom\" \/><\/sub>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Problem 13 \u2013 February 22, 2019 Some Are Not That Far Away Consider any five points P1, P2, P3, P4, and P5 in the interior of a square of side length 1. Denote by dij the distance between points Pi and Pj when i is not equal to j. Prove that at least one of [&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-239","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\/239","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=239"}],"version-history":[{"count":3,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/239\/revisions"}],"predecessor-version":[{"id":243,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/239\/revisions\/243"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}