{"id":92770,"date":"2017-12-23T03:26:53","date_gmt":"2017-12-23T05:26:53","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=92770"},"modified":"2020-11-03T15:01:50","modified_gmt":"2020-11-03T17:01:50","slug":"malabarismos-aritmeticos-e-algebricos-problemas-propostos","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/malabarismos-aritmeticos-e-algebricos-problemas-propostos\/","title":{"rendered":"Malabarismos aritm\u00e9ticos e alg\u00e9bricos &#8211; Problemas propostos"},"content":{"rendered":"<p style=\"margin:20px;\">\n<div class=\"gr1 cbch3\" style=\"line-height: 30px;padding-left:25px;\">\n<p style=\"margin:10px;\">\n<p class=\"ti1 cch\" style=\"line-height:30px;margin:10px\">Identidades Alg\u00e9bricas &#8211; Problemas propostos<\/p>\n<hr style=\"height:2px;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:40px 55px 30px 55px; line-height: 25px;\">\nMuitos dos problemas que ser\u00e3o apresentados a seguir s\u00e3o problemas de olimp\u00edadas e, portanto, podem exigir um pouco mais de racioc\u00ednio.<br \/>\nPara ajudar, apresentaremos abaixo  algumas identidades apresentadas no nosso Blog. <\/p>\n<div class=\"gr3 cbch1 bgch1\"  style=\"line-height: 30px; padding-left:35px; width:90%;\">\n[tex]ax+ay=a(x+y)[\/tex]<br \/>\n[tex]a^2-b^2=(a+b)(a-b)[\/tex]<br \/>\n[tex]a^2\\pm 2ab+b^2=(a\\pm b)^2[\/tex]<br \/>\n[tex]a^3\\pm 3a^2b+3ab^2\\pm b^3=(a\\pm b)^3[\/tex]<br \/>\n[tex]a^2+b^2+c^2+2ab+2ac+2bc=(a+b+c)^2[\/tex]<br \/>\n[tex]a^3+b^3+c^3+3(a+b)(a+c)(b+c)=(a+b+c)^3[\/tex]<br \/>\n[tex]x^2+x(a+b)+ab=(x+a)(x+b)[\/tex]<br \/>\n[tex]x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=(x+a)(x+b)(x+c)[\/tex]<br \/>\n[tex](ax+by)^2+(ay-bx)^2=(a^2+b^2)(x^2+y^2)[\/tex]<br \/>\n[tex]a^3\\pm b^3=(a\\pm b)(a^2\\mp ab+b^2)[\/tex]<br \/>\n[tex]a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)[\/tex]<br \/>\n[tex]a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\\dots+ab^{n-2}+b^{n-1})[\/tex], [tex]n[\/tex] natural<br \/>\n[tex]a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-\\dots-ab^{n-2}+b^{n-1})[\/tex], [tex]n[\/tex] \u00edmpar<br \/>\n[tex]1+2+3+ \\cdots+t=\\dfrac{(1+t)\\cdot t}{2}[\/tex]\n<\/div>\n<p style=\"margin:30px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73;  margin-top: 8px; margin-bottom: 8px;  border: 0 none\"\/>\n<hr style=\"height:4px; width:100%;background-color:#ae7c73;  margin-top: 8px; margin-bottom: 8px;  border: 0 none\"\/>\n<hr style=\"height:1px; width:100%;background-color:#ae7c73;  margin-top: 8px; margin-bottom: 8px;  border: 0 none\"\/>\n<p style=\"margin:30px;\">\n<p>[tex]1.[\/tex] Prove que se [tex]a+b+c=0[\/tex], ent\u00e3o:<\/p>\n<p style=\"margin:10px;\">\n<p>[tex](a) \\,  \\,  \\dfrac{a^5+b^5+c^5}{5}=\\dfrac{a^3+b^3+c^3}{3}\\times \\dfrac{a^2+b^2+c^2}{2},\\qquad \\qquad [\/tex] [tex]<br \/>\n(b) \\,  \\,  \\dfrac{a^7+b^7+c^7}{7}=\\dfrac{a^5+b^5+c^5}{5}\\times \\dfrac{a^2+b^2+c^2}{2}[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]2.[\/tex] (<strong>OCM<\/strong>) Se [tex]x^2+x+1=0[\/tex], calcule o valor num\u00e9rico de <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\left(x+\\dfrac{1}{x}\\right)^2+\\left(x^2+\\dfrac{1}{x^2}\\right)^2+\\left(x^3+\\dfrac{1}{x^3}\\right)^2+\\cdots+\\left(x^{27}+\\dfrac{1}{x^{27}}\\right)^2.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]3.[\/tex] (<strong>Torneio das Cidades<\/strong>) Sejam [tex]a[\/tex], [tex]b[\/tex], [tex]c[\/tex] e [tex]d[\/tex] n\u00fameros reais tais que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\quad \\qquad a^3+b^3+c^3+d^3=a+b+c+d=0.[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Prove que a soma de dois desses n\u00fameros \u00e9 zero.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]4.[\/tex] (<strong>Col\u00f4mbia<\/strong>) Sejam [tex]a[\/tex], [tex]b[\/tex], [tex]c[\/tex] reais tais que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad a^{12}+b^{12}+c^{12}=8\\quad [\/tex] e [tex]\\quad \\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{abc}=-\\dfrac{6}{a+b+c}.[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Calcule [tex] \\,  \\, a^6+b^6+c^6[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]5.[\/tex] (<strong>China<\/strong>) Seja [tex]\\alpha[\/tex] um n\u00famero real tal que [tex]\\alpha^3-\\alpha-1=0[\/tex]. Determine o valor num\u00e9rico de<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad\\sqrt[3]{3\\alpha^2 -4\\alpha}+\\alpha \\sqrt[4]{2\\alpha^2 +3\\alpha +2}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]6.[\/tex] (<strong>OMG<\/strong>) Seja [tex]a=2008[\/tex]. Determine o valor da soma <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\displaystyle\\sum_{k=-2007}^{k=2007}\\dfrac{1}{1+a^k}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]7.[\/tex] (<strong>REOIM<\/strong>) Demonstrar que<br \/>\n[tex]\\qquad \\qquad \\dfrac{1}{\\sqrt{2+\\sqrt{2}}}+\\dfrac{1}{\\sqrt{6+\\sqrt{6}}} \\gt \\dfrac{5}{6}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]8.[\/tex] (<strong>Canad\u00e1<\/strong>) Calcule:<br \/>\n[tex]\\quad(a)\\, \\sqrt{\\dfrac{1}{6}+\\dfrac{\\sqrt{5}}{18}}-\\sqrt{\\dfrac{1}{6}-\\dfrac{\\sqrt{5}}{18}}[\/tex];<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\quad (b)\\, \\sqrt{1+\\dfrac{2}{5}}\\cdot\\sqrt{1+\\dfrac{2}{6}}\\cdot\\sqrt{1+\\dfrac{2}{7}}\\cdot\\sqrt{1+\\dfrac{2}{8}}\\cdots \\sqrt{1+\\dfrac{2}{57}}\\cdot\\sqrt{1+\\dfrac{2}{58}}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]9.[\/tex] (<strong>Rom\u00eania<\/strong>) Sejam [tex]x \\, [\/tex] e [tex] \\, y[\/tex] n\u00fameros naturais tais que<br \/>\n[tex]\\quad 2^x.3^y=\\left(24^{\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\cdots+\\frac{1}{60}}\\right).\\left(24^{\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\cdots+\\frac{1}{60}}\\right)^2.\\left(24^{\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\cdots+\\frac{1}{60}}\\right)^3\\cdots \\left(24^{\\frac{1}{60}}\\right)^{59}.[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Determinar o valor de [tex]x+y[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]10.[\/tex] (<strong>R\u00fassia<\/strong>) Sejam [tex]a[\/tex] e [tex]b[\/tex] n\u00fameros reais n\u00e3o nulos que satisfazem \u00e0 equa\u00e7\u00e3o<br \/>\n[tex]\\qquad \\qquad a^2b^2(a^2b^2+4)=2(a^6+b^6).[\/tex]<br \/>\nMostre que [tex]a[\/tex] e [tex]b[\/tex] n\u00e3o podem ser ambos racionais.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]11.[\/tex] (<strong>REOIM<\/strong>) Dados os n\u00fameros<br \/>\n[tex]\\qquad \\alpha=\\sqrt{13}+\\sqrt{10+2\\sqrt{13}} \\quad[\/tex] e [tex]\\quad\\beta=\\sqrt{5+2\\sqrt{3}}+\\sqrt{18-2\\sqrt{3}+2\\sqrt{65-26\\sqrt{3}}}[\/tex],<br \/>\nmostrar que [tex]\\alpha=\\beta[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]12.[\/tex] (<strong>OCM<\/strong>) Determine qual \u00e9 o maior desses dois n\u00fameros:<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{123456+10^{999}}{123457+10^{999}} \\qquad [\/tex] ou  [tex]\\qquad \\dfrac{123457+10^{999}}{123458+10^{999}}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]13.[\/tex] (<strong><strong>Canad\u00e1<\/strong><\/strong>) Se os n\u00fameros reais positivos [tex]a_1, a_2, \\cdots a_n[\/tex] s\u00e3o os comprimentos dos lados de um pol\u00edgono inscrito em uma circunfer\u00eancia, tais que:<br \/>\n[tex]\\qquad \\qquad a_1^2+a_2^2+a_3^2+&#8230;+a_n^2=a_1a_2+a_2a_3+a_3a_4+&#8230;+a_na_1,[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>pode-se afirmar que o pol\u00edgono \u00e9 equil\u00e1tero? Justifique sua resposta.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]14.[\/tex] (<strong>AIME<\/strong>) Sejam [tex]x[\/tex] e [tex]y[\/tex] n\u00fameros reais, com [tex]x\\neq\\pm y[\/tex], tais que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\begin{cases}<br \/>\nx^3=13x+3y\\\\<br \/>\ny^3=13y+3x.\\\\<br \/>\n\\end{cases}[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Determine [tex](x^2-y^2)^2[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]15.[\/tex] (<strong>Canad\u00e1<\/strong>) Ache os n\u00fameros [tex]x,y\\in R_+[\/tex] tais que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\begin{cases}<br \/>\n\\sqrt{x}+2\\sqrt{y}=9\\\\<br \/>\nx-4y=9.\\\\<br \/>\n\\end{cases}[\/tex] <\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]16.[\/tex] (<strong>Alemanha<\/strong>) Sejam [tex]x[\/tex], [tex]y[\/tex], [tex]a[\/tex] e [tex]b[\/tex] reais tais que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\begin{cases}<br \/>\nx^2+y^2=1\\\\<br \/>\n\\dfrac{x^4}{a}+\\dfrac{y^4}{b}=\\dfrac{1}{a+b}\\\\<br \/>\n\\end{cases}.[\/tex]<\/p>\n<p style=\"margin:15px;\">\n<p>Prove que [tex]\\dfrac{x^8}{a^3}+\\dfrac{y^8}{b^3}=\\dfrac{1}{(a+b)^3}.[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]17.[\/tex] (<strong>Canad\u00e1<\/strong>) Determine todos os reais positivos [tex]x[\/tex], [tex]y[\/tex], [tex]z[\/tex] tais que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad  \\begin{cases}<br \/>\n\\dfrac{4x^2}{4x^2+1}=y\\\\<br \/>\n\\dfrac{4y^2}{4y^2+1}=z\\\\<br \/>\n\\dfrac{4z^2}{4z^2+1}=x.\\\\<br \/>\n\\end{cases}[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]18.[\/tex] (<strong>Stanford<\/strong>) Ache os n\u00fameros reais [tex]x[\/tex] e [tex]y[\/tex] tais que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad  \\begin{cases}<br \/>\n\\ x^4+2x^3-y=-\\dfrac{1}{4}+\\sqrt{3}\\\\<br \/>\n\\ y^4+2y^3-x=-\\dfrac{1}{4}-\\sqrt{3}.\\\\<br \/>\n\\end{cases}[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]19.[\/tex] (<strong>Cro\u00e1cia<\/strong>) Se [tex]ax^3=by^3=cz^3[\/tex] e [tex]\\dfrac{1}{x}+\\dfrac{1}{y}+\\dfrac{1}{z}=1[\/tex], prove que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\sqrt[3]{ax^2+by^2+cz^2}=\\sqrt[3]{a}+\\sqrt[3]{b}+\\sqrt[3]{c}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]20.[\/tex] Sejam [tex]x[\/tex], [tex]y[\/tex], [tex]z[\/tex] n\u00fameros reais n\u00e3o nulos tais que [tex]x+y+z=0[\/tex]. Prove que <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{x^2+y^2}{x+y}+\\dfrac{y^2+z^2}{y+z}+\\dfrac{x^2+z^2}{x+z}=\\dfrac{x^3}{yz}+\\dfrac{y^3}{xz}+\\dfrac{z^3}{xy}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]21.[\/tex] Qual a soma dos algarismos do n\u00famero natural n\u00e3o nulo  [tex]n[\/tex], de modo que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\sqrt{\\dfrac{25}{2}+\\sqrt{\\dfrac{625}{4}-n \\, } \\, } \\, + \\, \\sqrt{\\dfrac{25}{2}-\\sqrt{\\dfrac{625}{4}-n \\, } \\, }[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>seja  tamb\u00e9m um n\u00famero natural?<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]22.[\/tex] (<strong>Peru<\/strong>) Determine todos os inteiros positivos [tex]x[\/tex] tais que a express\u00e3o <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{\\sqrt{9^2+\\dfrac{1}{11^2}+\\dfrac{9^2}{100^2}}-\\dfrac{1}{1100}}{x+1}[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>defina um n\u00famero inteiro.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]23.[\/tex] (<strong>EUA<\/strong>) Determine o valor da express\u00e3o<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{1\\cdots}}}}[\/tex] .<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]24.[\/tex] (<em><strong>Sophie Germain<\/strong><\/em>) Fatore [tex]x^4+4y^4[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]25.[\/tex] Sem efetuar as distributividades e os produtos, desenvolva a express\u00e3o [tex](a+b+c+d)^3[\/tex], admitindo que [tex]a, \\, b, \\, c, \\, d \\, [\/tex] s\u00e3o n\u00fameros reais.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]26.[\/tex] (<strong>Mold\u00e1via<\/strong>)  Os n\u00fameros inteiros [tex]x[\/tex], [tex]y[\/tex], [tex]z[\/tex] satisfazem a rela\u00e7\u00e3o [tex]x+y+z=0[\/tex].<br \/>\nMostre que o n\u00famero [tex]2(x^4+y^4+z^4)[\/tex] \u00e9 um quadrado perfeito.<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]27.[\/tex] (<strong>Rioplatense<\/strong>) Ache o valor da soma <\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\quad \\sqrt{1+\\dfrac{1}{1^2}+\\dfrac{1}{2^2}}+\\sqrt{1+\\dfrac{1}{2^2}+\\dfrac{1}{3^2}}+\\sqrt{1+\\dfrac{1}{3^2}+\\dfrac{1}{4^2}}+&#8230;+\\sqrt{1+\\dfrac{1}{2005^2}+\\dfrac{1}{2006^2}}.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]28.[\/tex] (<strong>Espanha-Adaptado<\/strong>) Sejam [tex]a[\/tex], [tex]b[\/tex], [tex]c[\/tex] n\u00fameros reais n\u00e3o nulos com [tex]a+b+c\\neq 0[\/tex] tais que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad  \\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c}=\\dfrac{1}{a+b+c}[\/tex] .<\/p>\n<p style=\"margin:10px;\">\n<p>Mostre que, para [tex]n[\/tex] \u00edmpar, se verifica a igualdade<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{1}{a^n}+\\dfrac{1}{b^n}+\\dfrac{1}{c^n}=\\dfrac{1}{a^n+b^n+c^n}[\/tex] .<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]29.[\/tex] Sejam [tex]x[\/tex], [tex]y[\/tex], [tex]z[\/tex] n\u00fameros reais dois a dois distintos. Prove que a express\u00e3o<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{x(y+z)}{(x-y)(x-z)}+\\dfrac{y(x+z)}{(y-z)(y-x)}+\\dfrac{z(x+y)}{(z-x)(z-y)}[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>n\u00e3o depende de [tex]x[\/tex], [tex]y[\/tex] e [tex]z[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]30.[\/tex] Sejam [tex]x[\/tex], [tex]y[\/tex], [tex]z[\/tex] n\u00fameros reais distintos. Prove que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{x^2}{(x-y)(x-z)}+\\dfrac{y^2}{(y-z)(y-x)}+\\dfrac{z^2}{(z-x)(z-y)}=1.[\/tex]<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]31.[\/tex] (<strong>Treinamento Cone Sul<\/strong>) Sejam [tex]a, b, c, x, y, z[\/tex] reais distintos tais que [tex]ax+by+cz=0[\/tex]. Prove que<\/p>\n<p style=\"margin:10px;\">\n<p>[tex]\\qquad \\qquad \\dfrac{ax^2+by^2+cz^2}{bc(y-z)^2+ca(z-x)^2+ab(x-y)^2}[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>n\u00e3o depende de [tex]x[\/tex], nem de [tex]y[\/tex], nem de [tex]z[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#ae7c73; border: 0 none\"\/>\n<p style=\"margin:20px;\">\n<p>[tex]32.[\/tex](<strong>Canad\u00e1<\/strong>) Os dois menores lados de um tri\u00e2ngulo ret\u00e2ngulo, [tex]a[\/tex] e [tex]b[\/tex], satisfazem \u00e0 desigualdade<\/p>\n<p style=\"margin:20px;\">\n<p>[tex]\\qquad \\qquad \\sqrt{a^2-6a\\sqrt{2}+19}+\\sqrt{b^2-4b\\sqrt{3}+16}\\leq 3.[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Encontre o per\u00edmetro desse tri\u00e2ngulo.<\/p>\n<p style=\"margin:20px\">\n<hr style=\"height:1px;background-color:#ae7c73;   margin-top: 8px; margin-bottom: 8px; border: 0 none\"\/>\n<hr style=\"height:1px;background-color:#ae7c73;   margin-top: 8px; margin-bottom: 8px; border: 0 none\"\/>\n<p style=\"text-align: right;\"><span style=\"color: #ae7c73; font-size: 20px;\"><strong>Equipe COM &#8211; OBMEP<\/strong><\/span><\/p>\n<\/div>\n<p class=\"fim\"><a title=\"Voltar.\" href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=67077\" target=\"_blank\"><span style=\"color: #ae7c73;\"><strong>Voltar para a Sala 3<\/strong><\/span><\/a><\/p>\n<table style=\" height:auto; margin-left: auto;margin-right:auto;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"explicador_p\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"90\" height=\"108\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala bg33 cb33\" style=\"width:280px; line-height: 20px; font-size:16px\" > Se voc\u00ea n\u00e3o conseguiu resolver algum problema, clique <a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=90580\" target=\"_blank\"><strong>AQUI<\/strong><\/a>.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Identidades Alg\u00e9bricas &#8211; Problemas propostos Muitos dos problemas que ser\u00e3o apresentados a seguir s\u00e3o problemas de olimp\u00edadas e, portanto, podem exigir um pouco mais de racioc\u00ednio. Para ajudar, apresentaremos abaixo algumas identidades apresentadas no nosso Blog. [tex]ax+ay=a(x+y)[\/tex] [tex]a^2-b^2=(a+b)(a-b)[\/tex] [tex]a^2\\pm 2ab+b^2=(a\\pm b)^2[\/tex] [tex]a^3\\pm 3a^2b+3ab^2\\pm b^3=(a\\pm b)^3[\/tex] [tex]a^2+b^2+c^2+2ab+2ac+2bc=(a+b+c)^2[\/tex] [tex]a^3+b^3+c^3+3(a+b)(a+c)(b+c)=(a+b+c)^3[\/tex] [tex]x^2+x(a+b)+ab=(x+a)(x+b)[\/tex] [tex]x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=(x+a)(x+b)(x+c)[\/tex] [tex](ax+by)^2+(ay-bx)^2=(a^2+b^2)(x^2+y^2)[\/tex] [tex]a^3\\pm b^3=(a\\pm b)(a^2\\mp ab+b^2)[\/tex] [tex]a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)[\/tex] &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/malabarismos-aritmeticos-e-algebricos-problemas-propostos\/\">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-92770","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/92770","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=92770"}],"version-history":[{"count":15,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/92770\/revisions"}],"predecessor-version":[{"id":169270,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/92770\/revisions\/169270"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=92770"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}