{"id":128831,"date":"2019-06-05T20:32:49","date_gmt":"2019-06-05T23:32:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=128831"},"modified":"2024-06-10T22:26:55","modified_gmt":"2024-06-11T01:26:55","slug":"problema-para-ajudar-na-escola-vamos-contar","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-vamos-contar\/","title":{"rendered":".Problema para ajudar na escola: Vamos contar?"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220247\" 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: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 8\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: Dif\u00edcil<\/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>OME<\/strong>, 2003 &#8211; Adaptado) Quantas ternas ordenadas [tex](x, y, z)[\/tex] de n\u00fameros naturais maiores do que [tex]1[\/tex] s\u00e3o tais que [tex] \\, \\boxed{x \\cdot y \\cdot z=5^{30}}[\/tex]?<\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb36 g1\" style=\"margin:30px;line-height:27px; width:700px; 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=\"65\" height=\"77\" class=\"alignright size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\" style=\"line-height:20px; margin-bottom: 10px;\"> Ajuda <\/p>\n<p style=\"margin:20px\">\n<p>A soma dos [tex]t[\/tex] primeiros n\u00fameros naturais n\u00e3o nulos \u00e9 dada por<br \/>\n[tex]\\qquad \\qquad \\boxed{1+2+3+ \\cdots+t=\\dfrac{(1+t)\\cdot t}{2}}[\/tex]. <span style=\"font-size:11px;\">(Se precisar, visite <strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=85649\" target=\"_blank\" title=\"Sala para leitura: A soma 1 + 2 + 3 + ... + t.\" >esta p\u00e1gina<\/a><\/strong>.)<\/span>\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\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\"\/>\n<p>Sejam [tex]x, y, z[\/tex] n\u00fameros naturais maiores do que [tex]1[\/tex] tais que [tex] \\, \\boxed{x \\cdot y \\cdot z=5^{30}}.[\/tex]<br \/>\nComo [tex]5[\/tex] \u00e9 primo e [tex]x, y, z[\/tex] s\u00e3o n\u00fameros naturais maiores do que [tex]1[\/tex], ent\u00e3o:<\/p>\n<ul>\n<li>[tex]x \\cdot y \\cdot z=5^a \\cdot 5^b \\cdot 5^c=5^{a+b+c}=5^{30}[\/tex], com [tex] a,b,c[\/tex] n\u00fameros naturais n\u00e3o nulos.<\/li>\n<\/ul>\n<p>Dessa forma, o n\u00famero de  ternas ordenadas [tex](x, y, z)[\/tex] de n\u00fameros naturais maiores do que [tex]1[\/tex] tais que [tex]x \\cdot y \\cdot z=5^{30}[\/tex] \u00e9 igual ao n\u00famero de  ternas ordenadas [tex](a, b, c)[\/tex] de n\u00fameros naturais n\u00e3o nulos tais que [tex]a+b+c=30.[\/tex]<br \/>\nVamos ent\u00e3o fazer a contagem das ternas ordenadas [tex](a, b, c)[\/tex]; para isso, observe a tabela abaixo.<\/p>\n<table style=\"border: 2px solid #616161; width:300px; text-align:center; height:auto; margin-left:auto; margin-right:auto\">\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]a[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]b[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]c[\/tex]<\/td>\n<td  style=\"width:55%;\">N\u00famero de ternas<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]28[\/tex]<\/td>\n<td rowspan=\"6\" style=\"vertical-align:middle; border: 2px solid #616161;\">[tex]\\textcolor{red}{28}[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]27[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]26[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]\\cdots[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]27[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]2[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]28[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;border: 2px solid #616161;\">[tex]27[\/tex]<\/td>\n<td rowspan=\"6\" style=\"vertical-align:middle;;border: 2px solid #616161;\">[tex]\\textcolor{red}{27}[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]26[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]25[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]\\cdots[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]26[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]2[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]27[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;border: 2px solid #616161;\">[tex]26[\/tex]<\/td>\n<td rowspan=\"6\" style=\"vertical-align:middle;\">[tex]\\textcolor{red}{26}[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]25[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]24[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\cdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]\\cdots[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]25[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]2[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]26[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\vdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]\\vdots[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]\\vdots[\/tex]<\/td>\n<td  style=\"width:55%;\">[tex]\\vdots[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]27[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;border: 2px solid #616161;\">[tex]2[\/tex]<\/td>\n<td rowspan=\"2\" style=\"vertical-align:middle;;border: 2px solid #616161;\">[tex]\\textcolor{red}{2}[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]27[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1[\/tex]<\/td>\n<\/tr>\n<tr style=\"border: 2px solid #616161;width:15%;\">\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]28[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;width:15%;\">[tex]1[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1[\/tex]<\/td>\n<td  style=\"width:55%;\">[tex]\\textcolor{red}{1}[\/tex]<\/td>\n<\/tr>\n<\/table>\n<p>Analisando a tabela e utilizando a f\u00f3rmula do Lembrete, conclu\u00edmos que o n\u00famero de ternas que satisfazem as condi\u00e7\u00f5es do problema \u00e9 dado por:<br \/>\n[tex]\\qquad \\qquad 1+2+3+ \\cdots+27+28=\\dfrac{(1+28)\\cdot 28}{2}= \\, \\fcolorbox{black}{#eee0e5}{$406$} \\, .[\/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\/Vamos-contar.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Vamos contar.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 8\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: Dif\u00edcil) (OME, 2003 &#8211; Adaptado) Quantas ternas ordenadas [tex](x, y, z)[\/tex] de n\u00fameros naturais maiores do que [tex]1[\/tex] s\u00e3o tais que [tex] \\, \\boxed{x \\cdot y \\cdot z=5^{30}}[\/tex]? Ajuda A soma dos [tex]t[\/tex] primeiros &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-vamos-contar\/\">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-128831","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/128831","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=128831"}],"version-history":[{"count":13,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/128831\/revisions"}],"predecessor-version":[{"id":220249,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/128831\/revisions\/220249"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=128831"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}