{"id":3345,"date":"2023-10-14T14:37:59","date_gmt":"2023-10-14T18:37:59","guid":{"rendered":"https:\/\/josmfs.net\/?p=3345"},"modified":"2023-10-14T14:37:59","modified_gmt":"2023-10-14T18:37:59","slug":"amazing-identity","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2023\/10\/14\/amazing-identity\/","title":{"rendered":"Amazing Identity"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-3343\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/10\/Amazing-Identity-Fig2.jpg\" alt=\"\" width=\"200\" height=\"274\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/10\/Amazing-Identity-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/10\/Amazing-Identity-Fig2-219x300.jpg 219w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/>This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.<\/p>\n<p>\u201c31.\u00a0 Prove that if <em>n<\/em> is a natural number, then we have<\/p>\n<p style=\"text-align: center;\">(\u221a2 \u2013 1)<em><sup>n<\/sup><\/em> = \u221a<em>m<\/em> \u2013 \u221a(<em>m<\/em> \u2013 1),<\/p>\n<p>where <em>m<\/em> is a natural number.\u201d<\/p>\n<p>Here, natural numbers are 1, 2, 3, \u2026<\/p>\n<p>I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/10\/Amazing-Identity-210928.pdf\">Amazing Identity<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads. \u201c31.\u00a0 Prove that if n is a natural number, then we have (\u221a2 \u2013 1)n = \u221am \u2013 \u221a(m \u2013 1), where m is a natural number.\u201d Here, natural numbers are 1, 2, 3, \u2026 I found it to be quite [&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":[104,222],"class_list":["post-3345","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-mathematical-induction","tag-polish-math-olympiads"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3345","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=3345"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3345\/revisions"}],"predecessor-version":[{"id":3346,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3345\/revisions\/3346"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=3345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=3345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=3345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}