{"id":902,"date":"2020-05-28T11:08:24","date_gmt":"2020-05-28T11:08:24","guid":{"rendered":"http:\/\/blog.atlant.is\/?p=902"},"modified":"2021-02-21T12:11:56","modified_gmt":"2021-02-21T10:11:56","slug":"imo-2019-problem-1","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=902","title":{"rendered":"IMO 2019 Problem 1"},"content":{"rendered":"\n<p class=\"has-text-align-center\">An exercice with <a rel=\"noreferrer noopener\" href=\"https:\/\/www.mathjax.org\/\" target=\"_blank\">MathJax<\/a><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/www.imo-official.org\/\" target=\"_blank\" rel=\"noopener noreferrer\"><img decoding=\"async\" loading=\"lazy\" width=\"300\" height=\"206\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/05\/imo-logo.png\" alt=\"\" class=\"wp-image-949\"\/><\/a><figcaption><a href=\"https:\/\/www.imo-official.org\/\" target=\"_blank\" rel=\"noreferrer noopener\">Olympiade Internationale de Math\u00e9matiques<\/a><\/figcaption><\/figure><\/div>\n\n\n\n<p>Problem 1: Let $\\mathbb{Z}$ be the set of integers. Determine all functions $f : \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that<\/p>\n\n\n\n<p>\\begin{equation}<br>\\forall (a, b) \\in \\mathbb{Z} \\times \\mathbb{Z}, \\quad f(2a) + 2 f(b) = f(f(a + b))<br>\\label{eq1}<br>\\end{equation}<\/p>\n\n\n\n<!--more-->\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>Substituting $0$ for $a$, $x$ for $b$ in $(\\ref{eq1})$ yields:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\forall x \\in \\mathbb{Z}, \\quad f(0) + 2 f(x) = f(f(x))<br>\\label{eq2}<br>\\end{equation}<\/p>\n\n\n\n<p>Substituting $1$ for $a$, $x &#8211; 1$ for $b$ in $(\\ref{eq1})$ yields:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\forall x \\in \\mathbb{Z}, \\quad f(2) + 2 f(x &#8211; 1) = f(f(x))<br>\\label{eq3}<br>\\end{equation}<\/p>\n\n\n\n<p>Combining $(\\ref{eq2})$ and $(\\ref{eq3})$ yields:<\/p>\n\n\n\n<p>$$\\forall x \\in \\mathbb{Z}, \\quad f(x) = f(x &#8211; 1) + {{f(2) &#8211; f(0)} \\over 2}$$<\/p>\n\n\n\n<p>Thus $f$ is necessary of the form $f(x) = m x + n$ for some $(m, n) \\in  \\mathbb{Z} \\times \\mathbb{Z}$.<\/p>\n\n\n\n<p>Expanding $(\\ref{eq1})$ yields:<\/p>\n\n\n\n<p>$$\\forall (a, b) \\in \\mathbb{Z} \\times \\mathbb{Z}, \\quad m (2 &#8211; m)(a + b) + n(2 &#8211; m) = 0$$<\/p>\n\n\n\n<p>It follows that necessarily either $m = 2$ or $m = n = 0$.<\/p>\n\n\n\n<p>We can verify that these values actually give solutions to equation $(\\ref{eq1})$.<\/p>\n\n\n\n<p>Thus the set of functions $f$ solutions to problem 1 is exactly:<\/p>\n\n\n\n<p>$$\\{ f: x \\mapsto 0 \\} \\cup \\{f: x \\mapsto 2x + n, n \\in \\mathbb{Z} \\}$$<\/p>\n\n\n\n<p>$\\Box$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>An exercice with MathJax Problem 1: Let $\\mathbb{Z}$ be the set of integers. Determine all functions $f : \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that \\begin{equation}\\forall (a, b) \\in \\mathbb{Z} \\times \\mathbb{Z}, \\quad f(2a) + 2 f(b) = f(f(a + b))\\label{eq1}\\end{equation}<\/p>\n","protected":false},"author":1,"featured_media":949,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[19],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/902"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=902"}],"version-history":[{"count":46,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/902\/revisions"}],"predecessor-version":[{"id":3452,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/902\/revisions\/3452"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/949"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=902"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=902"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=902"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=902"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}