{"id":2621,"date":"2022-10-09T15:21:58","date_gmt":"2022-10-09T15:21:58","guid":{"rendered":"https:\/\/blogs.ua.es\/dimates\/?p=2621"},"modified":"2022-10-09T15:32:03","modified_gmt":"2022-10-09T15:32:03","slug":"solucion-a-el-cuadrado-misterioso","status":"publish","type":"post","link":"https:\/\/blogs.ua.es\/dimates\/2022\/10\/09\/solucion-a-el-cuadrado-misterioso\/","title":{"rendered":"Soluci\u00f3n a el cuadrado misterioso"},"content":{"rendered":"<pre>Problema 1 del nivel A de la Olimpiada Auton\u00f3mica de la Comunidad Valenciana\r\nSe dirige a una edad de: 12-13 a\u00f1os<\/pre>\n<p>Si (a + 1\/a)\u00b2 = 5, \u00bfcu\u00e1nto vale (a\u00b3 + 1\/a\u00b3)\u00b2?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2619\" src=\"https:\/\/blogs.ua.es\/dimates\/files\/2022\/10\/265.cuadrado.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/blogs.ua.es\/dimates\/files\/2022\/10\/265.cuadrado.png 300w, https:\/\/blogs.ua.es\/dimates\/files\/2022\/10\/265.cuadrado-150x150.png 150w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\nSoluci\u00f3n:<br \/>\n<!--more--><br \/>\nEmpecemos por desarrollar un poco la expresi\u00f3n (a + 1\/a)\u00b2 = 5. Si nos fijamos bien, resulta que da a\u00b2 + 2 + 1\/a\u00b2 = 5, es decir, a\u00b2 + 1\/a\u00b2 = 3.<\/p>\n<p>Para el resto de razonamiento, podemos suponer que a + 1\/a = ra\u00edz(5). Cambiar signos supondr\u00eda algunos cambios, pero no demasiados.<\/p>\n<p>Y esto puede dar una idea muy oportuna, ya que (a + 1\/a)\u00b3 se puede parecer a la expresi\u00f3n que queremos, que est\u00e1 dentro del cuadrado, ya que al multiplicar la a\u00b2 + 1\/a\u00b2 = 3 por a + 1\/a = ra\u00edz(5), obtenemos a\u00b3 + 1\/a\u00b3 + a + 1\/a = 3ra\u00edz(5).<\/p>\n<p>Puesto que a + 1\/a = +ra\u00edz(5), como (a + 1\/a)\u00b3 es  +-5\u00b7ra\u00edz(5), resulta que  a\u00b3 + 1\/a\u00b3 + ra\u00edz(5) = +3\u00b7ra\u00edz(5), es decir, a\u00b3 + 1\/a\u00b3 = +2\u00b7ra\u00edz(5).<\/p>\n<p>De esta forma (estudiando los dos posibles signos, tenemos que a\u00b3 + 1\/a\u00b3 = +-2\u00b7ra\u00edz(5)), (a\u00b3 + 1\/a\u00b3)\u00b2 = 4\u00b75 = 20.<\/p>\n<p>Otra manera mucho m\u00e1s dif\u00edcil de hacerlo es calcular (de forma aproximada) el valor de a y usar la calculadora para calcular el valor solicitado.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Problema 1 del nivel A de la Olimpiada Auton\u00f3mica de la Comunidad Valenciana Se dirige a una edad de: 12-13 a\u00f1os Si (a + 1\/a)\u00b2 = 5, \u00bfcu\u00e1nto vale (a\u00b3 + 1\/a\u00b3)\u00b2? Soluci\u00f3n:<\/p>\n","protected":false},"author":4267,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2242020,1738,2849,3303],"tags":[],"class_list":["post-2621","post","type-post","status-publish","format-standard","hentry","category-olimpiada-de-la-comunidad-valenciana","category-olimpiadas","category-problemas","category-soluciones"],"_links":{"self":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2621","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/users\/4267"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/comments?post=2621"}],"version-history":[{"count":3,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2621\/revisions"}],"predecessor-version":[{"id":2624,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2621\/revisions\/2624"}],"wp:attachment":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/media?parent=2621"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/categories?post=2621"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/tags?post=2621"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}