{"id":154674,"date":"2020-05-23T19:52:06","date_gmt":"2020-05-23T22:52:06","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=154674"},"modified":"2024-06-25T03:10:19","modified_gmt":"2024-06-25T06:10:19","slug":"problema-para-ajudar-na-escola-ali-babao-e-a-decima-quinta-de-suas-40-equacoes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-ali-babao-e-a-decima-quinta-de-suas-40-equacoes\/","title":{"rendered":".Problema para ajudar na escola: Ali Bab\u00e3o e a d\u00e9cima quinta de suas 40 equa\u00e7\u00f5es"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220827\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#A0522D;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#0000FF\"> (A partir do 2\u00ba ano do E. M. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: Muito Dif\u00edcil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 1px; margin-bottom: 20px;border: 0 none\"\/>\n<p style=\"margin:10px\">\n<p>(<strong>ONEM<\/strong>, 2011) Se [tex]x,\\, y[\/tex] e [tex]z[\/tex] s\u00e3o n\u00fameros reais tais que [tex]x \\geqslant 1[\/tex], [tex]y \\geqslant \\frac{1}{2}[\/tex] e [tex]z \\geqslant \\frac{3}{2}\\\\[\/tex], encontre <strong>TODAS <\/strong>as solu\u00e7\u00f5es da equa\u00e7\u00e3o<br \/>\n[tex]\\qquad \\qquad \\boxed{\\left(xyz\\right)^2=12\\left(x-1 \\right)\\left(2y-1 \\right)\\left(2z-3 \\right)}\\,.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 1px; margin-bottom: 20px;border: 0 none\"\/>\n<p style=\"margin:10px\">\n<p>Como do lado esquerdo da equa\u00e7\u00e3o dada temos o produto [tex]x^2y^2z^2[\/tex], iniciaremos a solu\u00e7\u00e3o deste problema tentando responder as seguintes perguntas:<\/p>\n<ul>\n<li>Como relacionar [tex]x^2[\/tex] a [tex]x-1[\/tex]?<\/li>\n<li>Como relacionar [tex]y^2[\/tex] a [tex]2y-1[\/tex]?<\/li>\n<li>Como relacionar [tex]z^2[\/tex] a [tex]2z-3[\/tex]?<\/li>\n<\/ul>\n<p>Vamos come\u00e7ar pela segunda pergunta, cuja resposta \u00e9 a mais intuitiva.<\/p>\n<ul>\n<li>Observe que [tex]\\left(y-1\\right)^2=y^2-2y+1[\/tex] e [tex]\\left(y-1\\right)^2 \\geqslant 0[\/tex].<\/li>\n<p>Assim:<br \/>\n[tex]\\qquad 0 \\leqslant\\left(y-1\\right)^2=y^2-2y+1[\/tex]<br \/>\n[tex]\\qquad \\textcolor{red}{2y-1} \\leqslant \\textcolor{red}{y^2} \\,. \\qquad \\textcolor{#800000}{(i)}[\/tex]<\/ul>\n<p>Vamos seguir pelo mesmo caminho e responder a terceira pergunta.<\/p>\n<ul>\n<li>Note que [tex]\\left(z-3\\right)^2=z^2-6z+9[\/tex] e [tex]\\left(z-3\\right)^2 \\geqslant 0[\/tex].<\/li>\n<p>Com isso:<br \/>\n[tex]\\qquad 0 \\leqslant\\left(z-3\\right)^2=z^2-6z+9[\/tex]<br \/>\n[tex]\\qquad 6z-9 \\leqslant z^2 [\/tex]<br \/>\n[tex]\\qquad 3\\textcolor{red}{\\left(2z-3\\right)} \\leqslant \\textcolor{red}{z^2} \\,. \\qquad \\textcolor{#800000}{(ii)} [\/tex]<\/ul>\n<ul>\n<li>Para finalizar, veja que [tex]\\left(x-2\\right)^2=x^2-4x+4[\/tex] e [tex]\\left(x-2\\right)^2 \\geqslant 0[\/tex].<\/li>\n<p>Logo:<br \/>\n[tex]\\qquad 0 \\leqslant\\left(x-2\\right)^2=x^2-4x+4[\/tex]<br \/>\n[tex]\\qquad 4x-4 \\leqslant x^2 [\/tex]<br \/>\n[tex]\\qquad 4\\textcolor{red}{\\left(x-1\\right)} \\leqslant \\textcolor{red}{x^2} \\,. \\qquad \\textcolor{#800000}{(iii)} [\/tex]<\/ul>\n<p>Por outro lado, por hip\u00f3tese temos que:<br \/>\n[tex]\\qquad x \\geqslant 1[\/tex], [tex]y \\geqslant \\dfrac{1}{2}\\, [\/tex] e [tex]\\, z \\geqslant \\dfrac{3}{2}\\\\[\/tex],<br \/>\ndonde segue que:<br \/>\n[tex]\\qquad x -1\\geqslant 0[\/tex], [tex]y-\\dfrac{1}{2}\\geqslant 0\\, [\/tex] e [tex]\\, z-\\dfrac{3}{2} \\geqslant 0 [\/tex].<br \/>\nCom isso,<br \/>\n[tex]\\qquad x -1\\geqslant 0[\/tex], [tex]\\dfrac{2y-1}{2}\\geqslant 0\\, [\/tex] e [tex]\\, \\dfrac{2z-3}{2} \\geqslant 0 [\/tex]<br \/>\ne, portanto,<br \/>\n[tex]\\qquad \\fcolorbox{red}{#ffffff}{$x -1\\geqslant 0$}\\;[\/tex], [tex]\\; \\fcolorbox{red}{#ffffff}{$2y-1 \\geqslant 0$}\\; [\/tex] e [tex]\\; \\fcolorbox{red}{#ffffff}{$2z-3 \\geqslant 0$}\\,. \\qquad \\textcolor{#800000}{(iv)}  [\/tex]<br \/>\nConsiderando as desigualdades obtidas em [tex]\\textcolor{#800000}{(iv)}[\/tex], podemos multiplicar as desigualdades [tex]\\textcolor{#800000}{(i)}[\/tex], [tex]\\textcolor{#800000}{(ii)}[\/tex] e [tex]\\textcolor{#800000}{(iii)}[\/tex] e obteremos:<br \/>\n[tex]\\qquad 4\\left(x-1\\right)\\cdot \\left(2y-1 \\right) \\cdot 3\\left(2z-3\\right) \\leqslant x^2\\cdot y^2 \\cdot z^2 [\/tex]<br \/>\n[tex]\\qquad 12\\left(x-1\\right) \\left(2y-1 \\right)  \\left(2z-3\\right) \\textcolor{red}{\\leqslant} \\left(xyz \\right)^2 [\/tex].<br \/>\nMas estamos interessados nos n\u00fameros reais [tex]x,\\, y[\/tex] e [tex]z[\/tex] que satisfazem a igualdade<br \/>\n[tex]\\qquad 12\\left(x-1 \\right)\\left(2y-1 \\right)\\left(2z-3 \\right)\\textcolor{red}{=}\\left(xyz\\right)^2[\/tex],<br \/>\ne essa igualdade ocorre somente quando em  [tex]\\textcolor{#800000}{(i)}[\/tex], [tex]\\textcolor{#800000}{(ii)}[\/tex] e [tex]\\textcolor{#800000}{(iii)}[\/tex] ocorrerem igualdades e isso s\u00f3 \u00e9 poss\u00edvel se:<\/p>\n<ul>\n<li>[tex]\\boxed{\\left(y-1\\right)^2=0}\\quad[\/tex] , [tex]\\quad \\boxed{\\left(z-3\\right)^2=0}\\quad [\/tex] e [tex]\\quad \\boxed{\\left(x-2\\right)^2=0}[\/tex]. <\/li>\n<\/ul>\n<p>Dessa forma, segue que<\/p>\n<ul>\n<li>[tex]\\boxed{y-1=0}\\quad[\/tex] , [tex]\\quad \\boxed{z-3=0}\\quad [\/tex] e [tex]\\quad \\boxed{x-2=0}[\/tex]. <\/li>\n<\/ul>\n<p>e, portanto, temos apenas uma solu\u00e7\u00e3o para a d\u00e9cima quinta equa\u00e7\u00e3o do mestre Ali Bab\u00e3o: [tex]\\fcolorbox{black}{#eee0e5}{$\\{x=2\\,;\\,y=1\\,;\\;z=3\\}$}\\,[\/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\/2020\/05\/Ali-Bab\u00e3o-e-a-d\u00e9cima-quinta-de-suas-40-equa\u00e7\u00f5es.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Ali Bab\u00e3o e a d\u00e9cima quinta de suas 40 equa\u00e7\u00f5es.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (A partir do 2\u00ba ano do E. M. &#8211; N\u00edvel de dificuldade: Muito Dif\u00edcil) (ONEM, 2011) Se [tex]x,\\, y[\/tex] e [tex]z[\/tex] s\u00e3o n\u00fameros reais tais que [tex]x \\geqslant 1[\/tex], [tex]y \\geqslant \\frac{1}{2}[\/tex] e [tex]z \\geqslant \\frac{3}{2}\\\\[\/tex], encontre TODAS as solu\u00e7\u00f5es da equa\u00e7\u00e3o [tex]\\qquad \\qquad \\boxed{\\left(xyz\\right)^2=12\\left(x-1 \\right)\\left(2y-1 &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-ali-babao-e-a-decima-quinta-de-suas-40-equacoes\/\">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-154674","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/154674","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=154674"}],"version-history":[{"count":8,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/154674\/revisions"}],"predecessor-version":[{"id":220829,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/154674\/revisions\/220829"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=154674"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}