{"id":138168,"date":"2019-10-15T22:43:02","date_gmt":"2019-10-16T01:43:02","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=138168"},"modified":"2024-06-22T00:33:05","modified_gmt":"2024-06-22T03:33:05","slug":"problema-para-ajudar-na-escola-quantas-quintas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-quantas-quintas\/","title":{"rendered":".Problema para ajudar na escola: Quantas quintas?"},"content":{"rendered":"

✏ Link do problema para dispositivos da Apple<\/em>.<\/span><\/a><\/strong><\/p>\n

\n

Problema
\n (A partir do 6\u00ba ano do E. F. – N\u00edvel de dificuldade: M\u00e9dio<\/span>)<\/span><\/p>\n

\n(ONEM<\/strong>, 2010) Em um determinado ano, um m\u00eas espec\u00edfico teve cinco sextas-feiras e cinco domingos. Quantas quintas-feiras teve o m\u00eas em quest\u00e3o?<\/p>\n

\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o <\/p>\n

\n
    \n
  • Vamos supor inicialmente que o primeiro domingo do m\u00eas em quest\u00e3o tivesse ocorrido na primeira semana do m\u00eas. Como ocorreram cinco domingos no m\u00eas, a distribui\u00e7\u00e3o m\u00ednima dos dias da semana nesse m\u00eas seria a mostrada na tabela abaixo.<\/li>\n

    [tex]\\begin{array}{r|r|r|r|r|r|r|r}
    \n&\\text{domingo} &\\text{segunda} & \\text{ ter\u00e7a }\\,&\\text{quarta} &\\text{quinta} &\\text{ sexta }&\\text{s\u00e1bado}\\\\
    \n\\hline
    \n\\text{1\u00aa semana}& \\textcolor{blue}{*}\\,\\,\\,^{\\boxed{1}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{2}} & \\textcolor{red}{*}\\,\\,\\,^{\\boxed{3}}& \\textcolor{red}{*}\\,\\,\\,^{\\boxed{4}}&\\textcolor{red}{*}\\,\\,\\,^{\\boxed{5}} &\\textcolor{#58E817}{*}\\,\\,\\,^{\\boxed{6}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{7}}\\\\
    \n\\hline
    \n\\text{2\u00aa semana}& \\textcolor{blue}{*}\\,\\,\\,^{\\boxed{8}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{9}} & \\textcolor{red}{*}^{\\boxed{10}}& \\textcolor{red}{*}^{\\boxed{11}}&\\textcolor{red}{*}^{\\boxed{12}} &\\textcolor{#58E817}{*}^{\\boxed{13}} &\\textcolor{red}{*}^{\\boxed{14}}\\\\
    \n\\hline
    \n\\text{3\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{15}} &\\textcolor{red}{*}^{\\boxed{16}} & \\textcolor{red}{*}^{\\boxed{17}}& \\textcolor{red}{*}^{\\boxed{18}}&\\textcolor{red}{*}^{\\boxed{19}} &\\textcolor{#58E817}{*}^{\\boxed{20}} &\\textcolor{red}{*}^{\\boxed{21}}\\\\
    \n\\hline
    \n\\text{4\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{22}} &\\textcolor{red}{*}^{\\boxed{23}} & \\textcolor{red}{*}^{\\boxed{24}}& \\textcolor{red}{*}^{\\boxed{25}}&\\textcolor{red}{*}^{\\boxed{26}} &\\textcolor{#58E817}{*}^{\\boxed{27}} &\\textcolor{red}{*}^{\\boxed{28}}\\\\
    \n\\hline
    \n\\text{5\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{29}} & & & & & &\\\\
    \n\\hline\\end{array}[\/tex]<\/p>\n

    Observe que, mesmo que o m\u00eas tivesse [tex]31[\/tex] dias, n\u00e3o teriam ocorrido cinco sextas-feiras nesse m\u00eas.<\/p>\n

    [tex]\\begin{array}{r|r|r|r|r|r|r|r}
    \n&\\text{domingo} &\\text{segunda} &\\,\\text{ter\u00e7a}\\, &\\text{quarta} &\\text{quinta} &\\text{ sexta }&\\text{s\u00e1bado}\\\\
    \n\\hline
    \n\\text{1\u00aa semana}& \\textcolor{blue}{*}\\,\\,\\,^{\\boxed{1}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{2}} & \\textcolor{red}{*}\\,\\,\\,^{\\boxed{3}}& \\textcolor{red}{*}\\,\\,\\,^{\\boxed{4}}&\\textcolor{red}{*}\\,\\,\\,^{\\boxed{5}} &\\textcolor{#58E817}{*}\\,\\,\\,^{\\boxed{6}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{7}}\\\\
    \n\\hline
    \n\\text{2\u00aa semana}& \\textcolor{blue}{*}\\,\\,\\,^{\\boxed{8}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{9}} & \\textcolor{red}{*}^{\\boxed{10}}& \\textcolor{red}{*}^{\\boxed{11}}&\\textcolor{red}{*}^{\\boxed{12}} &\\textcolor{#58E817}{*}^{\\boxed{13}} &\\textcolor{red}{*}^{\\boxed{14}}\\\\
    \n\\hline
    \n\\text{3\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{15}} &\\textcolor{red}{*}^{\\boxed{16}} & \\textcolor{red}{*}^{\\boxed{17}}& \\textcolor{red}{*}^{\\boxed{18}}&\\textcolor{red}{*}^{\\boxed{19}} &\\textcolor{#58E817}{*}^{\\boxed{20}} &\\textcolor{red}{*}^{\\boxed{21}}\\\\
    \n\\hline
    \n\\text{4\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{22}} &\\textcolor{red}{*}^{\\boxed{23}} & \\textcolor{red}{*}^{\\boxed{24}}& \\textcolor{red}{*}^{\\boxed{25}}&\\textcolor{red}{*}^{\\boxed{26}} &\\textcolor{#58E817}{*}^{\\boxed{27}} &\\textcolor{red}{*}^{\\boxed{28}}\\\\
    \n\\hline
    \n\\text{5\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{29}} & \\textcolor{red}{?}^{\\boxed{30}} & \\textcolor{red}{?}^{\\boxed{31}} & & & &\\\\
    \n\\hline\\end{array}\\\\
    \n\\,\\\\[\/tex]<\/p>\n

    Assim, o primeiro domingo do m\u00eas em quest\u00e3o n\u00e3o ocorreu na primeira semana do m\u00eas.<\/strong> <\/p>\n

  • Vamos supor agora que o primeiro domingo do m\u00eas em quest\u00e3o tivesse ocorrido na segunda semana do m\u00eas. Como ocorreram cinco domingos no m\u00eas, a distribui\u00e7\u00e3o m\u00ednima dos dias da semana nesse m\u00eas seria a mostrada na tabela abaixo.<\/li>\n

    \n[tex]\\begin{array}{c|c|c|c|c|c|c|c}
    \n&\\text{domingo} &\\text{segunda} & \\text{ ter\u00e7a } &\\text{quarta} &\\text{quinta} &\\text{ sexta }&\\text{s\u00e1bado}\\\\
    \n\\hline
    \n\\text{1\u00aa semana}& & & & & & &\\\\
    \n\\hline
    \n\\text{2\u00aa semana}& \\textcolor{blue}{*} & \\textcolor{red}{*} & \\textcolor{red}{*}& \\textcolor{red}{*}& \\textcolor{red}{*} & \\textcolor{#58E817}{*} & \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{3\u00aa semana}&\\textcolor{blue}{*} & \\textcolor{red}{*} & \\textcolor{red}{*}& \\textcolor{red}{*}& \\textcolor{red}{*} & \\textcolor{#58E817}{*} & \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{4\u00aa semana}& \\textcolor{blue}{*} & \\textcolor{red}{*} & \\textcolor{red}{*}& \\textcolor{red}{*}& \\textcolor{red}{*} & \\textcolor{#58E817}{*} & \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{5\u00aa semana}& \\textcolor{blue}{*} & \\textcolor{red}{*} & \\textcolor{red}{*}& \\textcolor{red}{*}& \\textcolor{red}{*} & \\textcolor{#58E817}{*} & \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{6\u00aa semana}& \\textcolor{blue}{*} & & & & & &\\\\
    \n\\hline\\end{array}[\/tex]<\/p>\n

    Como ocorreram cinco sextas-feiras, o m\u00eas teria [tex]31[\/tex] dias e a distribui\u00e7\u00e3o dos dias da semana nesse m\u00eas seria a mostrada na tabela abaixo.<\/p>\n

    [tex]\\begin{array}{r|r|r|r|r|r|r|r}
    \n&\\text{domingo} &\\text{segunda} & \\text{ ter\u00e7a } &\\text{quarta} &\\text{quinta} &\\text{ sexta }&\\text{s\u00e1bado}\\\\
    \n\\hline
    \n\\text{1\u00aa semana}& & & & & & \\textcolor{#58E817}{*}\\,\\,\\,^{\\boxed{1}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{2}} \\\\
    \n\\hline
    \n\\text{2\u00aa semana}& \\textcolor{blue}{*}\\,\\,\\,^{\\boxed{3}}& \\textcolor{red}{*}\\,\\,\\,^{\\boxed{4}}&\\textcolor{red}{*}\\,\\,\\,^{\\boxed{5}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{6}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{7}}&\\textcolor{#58E817}{*}\\,\\,\\,^{\\boxed{8}} &\\textcolor{red}{*}\\,\\,\\,^{\\boxed{9}} \\\\
    \n\\hline
    \n\\text{3\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{10}}& \\textcolor{red}{*}^{\\boxed{11}}&\\textcolor{red}{*}^{\\boxed{12}} &\\textcolor{red}{*}^{\\boxed{13}} &\\textcolor{red}{*}^{\\boxed{14}}& \\textcolor{#58E817}{*}^{\\boxed{15}} &\\textcolor{red}{*}^{\\boxed{16}}\\\\
    \n\\hline
    \n\\text{4\u00aa semana} & \\textcolor{blue}{*}^{\\boxed{17}}& \\textcolor{red}{*}^{\\boxed{18}}&\\textcolor{red}{*}^{\\boxed{19}} &\\textcolor{red}{*}^{\\boxed{20}} &\\textcolor{red}{*}^{\\boxed{21}}&\\textcolor{#58E817}{*}^{\\boxed{22}}&\\textcolor{red}{*}^{\\boxed{23}}\\\\
    \n\\hline
    \n\\text{5\u00aa semana}& \\textcolor{blue}{*}^{\\boxed{24}}& \\textcolor{red}{*}^{\\boxed{25}}&\\textcolor{red}{*}^{\\boxed{26}} &\\textcolor{red}{*}^{\\boxed{27}} &\\textcolor{red}{*}^{\\boxed{28}}& \\textcolor{#58E817}{*}^{\\boxed{29}} & \\textcolor{red}{*}^{\\boxed{30}}\\\\
    \n\\hline
    \n\\text{6\u00aa semana} & \\textcolor{blue}{*}^{\\boxed{31}} & & & &\\\\
    \n\\hline\\end{array}\\\\
    \n\\,\\\\[\/tex]<\/p>\n

    Neste caso, o m\u00eas teria [tex]31[\/tex] dias e teriam ocorrido quatro quintas-feiras nesse m\u00eas.<\/strong><\/p>\n

  • Finalmente, observe que n\u00e3o podemos ter o primeiro domingo do m\u00eas ocorrendo a partir da segunda semana: de fato, devemos ter pelo menos um dia na primeira semana, o que implica em termos a segunda semana cheia e, consequentemente, um domingo estaria nessa semana.<\/li>\n

    \n[tex]\\begin{array}{c|c|c|c|c|c|c|c}
    \n&\\text{domingo} &\\text{segunda} & \\text{ ter\u00e7a } &\\text{quarta} &\\text{quinta} &\\text{ sexta }&\\text{s\u00e1bado}\\\\
    \n\\hline
    \n\\text{1\u00aa semana}& \\textcolor{red}{?}& \\textcolor{red}{?}& \\textcolor{red}{?}& \\textcolor{red}{?}& \\textcolor{red}{?}& \\textcolor{red}{?}& \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{2\u00aa semana}&\\textcolor{red}{*} &\\textcolor{red}{*} &\\textcolor{red}{*} &\\textcolor{red}{*} &\\textcolor{red}{*} &\\textcolor{red}{*} &\\textcolor{red}{*}\\\\
    \n\\hline
    \n\\text{3\u00aa semana}& \\textcolor{blue}{*} & \\textcolor{red}{*} & \\textcolor{red}{*}& \\textcolor{red}{*}& \\textcolor{red}{*} & \\textcolor{#58E817}{*} & \\textcolor{red}{*}\\\\
    \n\\hline
    \n\\end{array}[\/tex]
    \n[tex]\\dots[\/tex]<\/p>\n<\/ul>\n

    Dessa forma, temos uma \u00fanica situa\u00e7\u00e3o que atende \u00e0s hip\u00f3teses do problema. E nesse caso, conforme exposto, temos um m\u00eas com [tex]31[\/tex] dias e com [tex]\\fcolorbox{black}{#eee0e5}{$ \\text{quatro quintas-feiras}$}.[\/tex]<\/p>\n

    \n

    Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n

    \n\n\n
    \"\"<\/td>\n\n

    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.
    \nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (Chrome<\/em>, Edge<\/em>, Firefox<\/em>, Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito Adobe Acrobat Reader<\/strong><\/em>.
    \nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n

    \n

    \nArquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    ✏ Link do problema para dispositivos da Apple. Problema (A partir do 6\u00ba ano do E. F. – N\u00edvel de dificuldade: M\u00e9dio) (ONEM, 2010) Em um determinado ano, um m\u00eas espec\u00edfico teve cinco sextas-feiras e cinco domingos. Quantas quintas-feiras teve o m\u00eas em quest\u00e3o? Solu\u00e7\u00e3o Vamos supor inicialmente que o primeiro domingo do m\u00eas em … <\/p>\n

    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-138168","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/138168","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=138168"}],"version-history":[{"count":13,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/138168\/revisions"}],"predecessor-version":[{"id":220568,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/138168\/revisions\/220568"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=138168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}