{"id":220619,"date":"2024-06-22T01:19:20","date_gmt":"2024-06-22T04:19:20","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=220619"},"modified":"2024-06-22T01:19:20","modified_gmt":"2024-06-22T04:19:20","slug":"a-problema-para-ajudar-na-escola-um-numero-natural-disfarcado","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/a-problema-para-ajudar-na-escola-um-numero-natural-disfarcado\/","title":{"rendered":"(A) Problema para ajudar na escola: Um n\u00famero natural disfar\u00e7ado&#8230;"},"content":{"rendered":"<p class=\"ti3\"  style=\"text-align:center;\">Clique no bot\u00e3o abaixo para visualizar o problema.<\/p>\n<div style=\"text-align:center;margin-top:10px;\">\n<input class=\"bo1 c1 cb13\" style=\"width:150px;\" value=\"Problema\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Problema';}\" \/>  <\/p>\n<div style=\"display: none;\">\n<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:20px; margin-bottom: 10px;\">Problema<br \/>\n<span style=\"font-size:12px;color:#00BF00\"> (A partir do 9\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: F\u00e1cil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>(<strong>ONEM<\/strong>, 2004) Para [tex]\\boxed{a=\\sqrt{23}+ \\sqrt{29}}\\,[\/tex] e [tex]\\,\\boxed{b=\\sqrt{23}- \\sqrt{29}}\\,[\/tex] a  soma<\/p>\n<p style=\"text-align:center\">[tex]S=\\dfrac{\\sqrt{23}\\left(ab+10 \\right)}{a+b}+\\dfrac{b+2\\sqrt{29}}{a}[\/tex]<\/p>\n<p>\u00e9 um n\u00famero natural.<br \/>\n<strong>Que n\u00famero \u00e9 esse?<\/strong><\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb36 g1\" style=\"margin:30px;line-height:30px; width:60%; margin-left:auto; margin-right:auto; \"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"80\" height=\"97\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\" style=\"line-height:20px; margin-bottom: 10px;\"> Lembrete<\/p>\n<p>Diferen\u00e7a de dois quadrados:<br \/>\n[tex]\\qquad \\qquad \\boxed{a^2-b^2=(a+b) \\cdot (a-b)}[\/tex]<br \/>\npara quaisquer [tex]\\, a,b\\in\\mathbb{R}.[\/tex]\n<\/p><\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:33px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nPara facilitar os c\u00e1lculos, observe que:<\/p>\n<ul>\n<li>[tex]a+b=\\left(\\sqrt{23}+ \\sqrt{29}\\right)+ \\left(\\sqrt{23}- \\sqrt{29}\\right)=2\\sqrt{23}[\/tex]<\/li>\n<\/ul>\n<p>e, pelo <span class=\"c3\" > <strong>Lembrete<\/strong><\/span>,<\/p>\n<ul>\n<li>[tex]ab=\\left(\\sqrt{23}+ \\sqrt{29}\\right)\\cdot \\left(\\sqrt{23}- \\sqrt{29}\\right)=\\left(\\sqrt{23}\\right)^2- \\left(\\sqrt{29}\\right)^2=23-29=-6.[\/tex]<\/li>\n<\/ul>\n<p>Assim, segue que:<br \/>\n[tex]\\qquad S=\\dfrac{\\sqrt{23}\\left(ab+10 \\right)}{a+b}+\\dfrac{b+2\\sqrt{29}}{a}\\\\<br \/>\n\\qquad S=\\dfrac{\\cancel{\\sqrt{23}}\\left(-6+10 \\right)}{2\\,\\cancel{\\sqrt{23}}}+\\dfrac{\\left(\\sqrt{23}- \\sqrt{29}\\right)+2\\sqrt{29}}{\\sqrt{23}+ \\sqrt{29}}\\\\<br \/>\n\\qquad S=\\dfrac{4}{2}+\\dfrac{\\sqrt{23}+\\sqrt{29}}{\\sqrt{23}+ \\sqrt{29}}\\\\<br \/>\n\\qquad S=2+1=3.[\/tex]<br \/>\nPortanto,  [tex]\\fcolorbox{black}{#eee0e5}{$S=\\dfrac{\\sqrt{23}\\left(ab+10 \\right)}{a+b}+\\dfrac{b+2\\sqrt{29}}{a}=3$}[\/tex] .<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:640px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"80\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:520px;background-color: #FFF5EE; padding:10px; font-family: calibri; font-size: 13px;color: #1C1C1C; line-height: 20px; \"> Se for conveniente, voc\u00ea pode obter um arquivo PDF desta p\u00e1gina, com o problema e a solu\u00e7\u00e3o, clicando no bot\u00e3o abaixo.<br \/>\nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (<em>Chrome<\/em>, <em>Edge<\/em>, <em>Firefox<\/em>, <em>Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito <em><strong>Adobe Acrobat Reader<\/strong><\/em>.<br \/>\nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o <strong><em>Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar <a href=\"http:\/\/get.adobe.com\/br\/reader\/\" rel=\"noopener\" target=\"_blank\"><strong>AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 100px;\">\n<p style=\"text-align: center; border-style: dotted; border-color :#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #800000;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/Um-n\u00famero-natural-disfar\u00e7ado.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Um n\u00famero natural disfar\u00e7ado.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Clique no bot\u00e3o abaixo para visualizar o problema. Problema (A partir do 9\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: F\u00e1cil) (ONEM, 2004) Para [tex]\\boxed{a=\\sqrt{23}+ \\sqrt{29}}\\,[\/tex] e [tex]\\,\\boxed{b=\\sqrt{23}- \\sqrt{29}}\\,[\/tex] a soma [tex]S=\\dfrac{\\sqrt{23}\\left(ab+10 \\right)}{a+b}+\\dfrac{b+2\\sqrt{29}}{a}[\/tex] \u00e9 um n\u00famero natural. Que n\u00famero \u00e9 esse? Lembrete Diferen\u00e7a de dois quadrados: [tex]\\qquad \\qquad \\boxed{a^2-b^2=(a+b) \\cdot (a-b)}[\/tex] para quaisquer [tex]\\, &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/a-problema-para-ajudar-na-escola-um-numero-natural-disfarcado\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-220619","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220619","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=220619"}],"version-history":[{"count":1,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220619\/revisions"}],"predecessor-version":[{"id":220620,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/220619\/revisions\/220620"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=220619"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}