{"id":57939,"date":"2016-03-17T20:41:01","date_gmt":"2016-03-17T23:41:01","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=57939"},"modified":"2023-03-31T20:41:55","modified_gmt":"2023-03-31T23:41:55","slug":"problema-de-olimpiada-presente-de-grego","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-de-olimpiada-presente-de-grego\/","title":{"rendered":".Problema de Olimp\u00edada:  Presente de Grego"},"content":{"rendered":"<p style=\"margin:20px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Problema<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n Determine todos os ternos de n\u00fameros reais [tex](a, b, c)[\/tex] satisfazendo \u00e0 equa\u00e7\u00e3o<\/p>\n<p style=\"text-align:center;\">\n[tex]a(a-b-c)+(b^2+c^2-bc)=4a^2\\left(abc-\\dfrac{a^2}{4}-b^2c^2\\right)[\/tex].<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nA equa\u00e7\u00e3o pode ser reescrita como sendo [tex]a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)[\/tex].<br \/>\nVeja, tamb\u00e9m, que: <\/p>\n<ul>\n<li>[tex]a^2+b^2+c^2-ab-ac-bc=\\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{2}\\geq 0[\/tex];<\/li>\n<li> [tex]a^2(4abc-a^2-4b^2c^2)=-a^2(a-2bc)^2\\leq 0[\/tex];<\/li>\n<\/ul>\n<p>assim<\/p>\n<ul>\n<li>[tex]0 \\le a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)\\le 0[\/tex].<\/li>\n<\/ul>\n<p>ou seja,<\/p>\n<ul>\n<li>[tex] a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)= 0[\/tex].<\/li>\n<\/ul>\n<p>Bom, como<br \/>\n[tex]\\quad a^2+b^2+c^2-ab-ac-bc=\\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{2}= 0[\/tex],<br \/>\nent\u00e3o<br \/>\n[tex]\\quad (a-b)^2+(b-c)^2+(c-a)^2= 0[\/tex].<br \/>\nMas, essa igualdade \u00e9 poss\u00edvel se, e somente se,  [tex](a-b)=(b-c)=(c-a)= 0[\/tex] e, portanto, temos  [tex]a=b=c[\/tex].<br \/>\nPor outro lado,<br \/>\n[tex]\\quad  a^2(4abc-a^2-4b^2c^2)=-a^2(a-2bc)^2= 0[\/tex],<br \/>\nent\u00e3o [tex]a=0[\/tex] ou [tex]a-2bc=0[\/tex].<br \/>\nResumindo, temos que  &#8220;[tex]a=b=c[\/tex]&#8221; e &#8220;[tex]a=0[\/tex] ou [tex]a-2bc=0[\/tex]&#8221;. E o que isso significa?<br \/>\n&#8211; Significa que<br \/>\n[tex]\\qquad a=b=c[\/tex] e [tex]a=0; \\qquad\\qquad  (i)[\/tex]<br \/>\nou<br \/>\n[tex]\\qquad a=b=c[\/tex] e  [tex]a-2bc=0. \\qquad\\qquad (ii)[\/tex]<br \/>\nDe [tex](i)[\/tex] tiramos que [tex]a=b=c=0[\/tex]. Se [tex]a=b=c\\ne 0[\/tex], de [tex](ii)[\/tex] conclu\u00edmos que [tex]a=b=c=\\dfrac{1}{2}[\/tex].<br \/>\nLogo, s\u00e3o dois os ternos poss\u00edveis: [tex](0, 0, 0)[\/tex] e  [tex]\\left(\\dfrac{1}{2}, \\dfrac{1}{2}, \\dfrac{1}{2}\\right)[\/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\" >Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"gg\">\nOlimp\u00edada: <strong>ONM<\/strong> &#8211; Gr\u00e9cia<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema Determine todos os ternos de n\u00fameros reais [tex](a, b, c)[\/tex] satisfazendo \u00e0 equa\u00e7\u00e3o [tex]a(a-b-c)+(b^2+c^2-bc)=4a^2\\left(abc-\\dfrac{a^2}{4}-b^2c^2\\right)[\/tex]. Solu\u00e7\u00e3o A equa\u00e7\u00e3o pode ser reescrita como sendo [tex]a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)[\/tex]. Veja, tamb\u00e9m, que: [tex]a^2+b^2+c^2-ab-ac-bc=\\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{2}\\geq 0[\/tex]; [tex]a^2(4abc-a^2-4b^2c^2)=-a^2(a-2bc)^2\\leq 0[\/tex]; assim [tex]0 \\le a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)\\le 0[\/tex]. ou seja, [tex] a^2+b^2+c^2-ab-ac-bc=a^2(4abc-a^2-4b^2c^2)= 0[\/tex]. Bom, como [tex]\\quad a^2+b^2+c^2-ab-ac-bc=\\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{2}= 0[\/tex], ent\u00e3o [tex]\\quad (a-b)^2+(b-c)^2+(c-a)^2= 0[\/tex]. Mas, essa igualdade \u00e9 poss\u00edvel &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-de-olimpiada-presente-de-grego\/\">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-57939","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/57939","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=57939"}],"version-history":[{"count":25,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/57939\/revisions"}],"predecessor-version":[{"id":185972,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/57939\/revisions\/185972"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=57939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}