{"id":6092,"date":"2021-02-22T06:00:07","date_gmt":"2021-02-22T04:00:07","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6092"},"modified":"2021-03-01T10:57:33","modified_gmt":"2021-03-01T08:57:33","slug":"sumas-de-potencias-que-son-cuadrados-perfectos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/sumas-de-potencias-que-son-cuadrados-perfectos\/","title":{"rendered":"Sumas de potencias que son cuadrados perfectos"},"content":{"rendered":"<p>El 2\u00ba problema de la ONEM (Olimp\u00edada Nacional Escolar de Matem\u00e1tica) de Mayo del 2020 dice<\/p>\n<blockquote><p>\n  Determinar si existen enteros positivos a, b y c, no necesariamente distintos, tales que <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=a%2Bb%2Bc%3D2020&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"a+b+c=2020\" class=\"latex\" \/> y <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=2%5Ea+%2B+2%5Eb+%2B+2%5Ec&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"2^a + 2^b + 2^c\" class=\"latex\" \/> es un cuadrado perfecto.\n<\/p><\/blockquote>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   soluciones :: Integer -> Integer -> [(Integer,Integer,Integer)]\n<\/pre>\n<p>tales que (soluciones k n) es la lista de las ternas no decrecientes (a,b,c) tales que que a+b+c=n y k^a + k^b + k^c es un cuadrado perfecto. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   soluciones 2 19  ==  [(2,6,11),(2,7,10),(4,7,8),(5,5,9),(6,6,7)]\n   soluciones 3 19  ==  []\n   take 2 (soluciones 2 2020)  ==  [(2,674,1344),(4,674,1342)]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nsoluciones :: Integer -> Integer -> [(Integer,Integer,Integer)]\nsoluciones k n =\n  [(a,b,c) | a <- [1..n `div` 3],\n             b <- [a..n-a],\n             let c = n-a-b,\n             c >= b,\n             esCuadradoPerfecto (k^a + k^b + k^c)]\n\n-- (esCuadradoPerfecto x) se verifica si x es un cuadrado perfecto. Por\n-- ejemplo,\n--    esCuadradoPerfecto 16  ==  True\n--    esCuadradoPerfecto 27  ==  False\nesCuadradoPerfecto :: Integer -> Bool\nesCuadradoPerfecto x =\n  (raizEntera x)^2 == x\n\n-- (raizEntera x) es el mayor entero cuyo cuadrado es menor o igual que\n-- x. Por ejemplo,\n--    raizEntera 16  ==  4\n--    raizEntera 27  ==  5\nraizEntera :: Integer -> Integer\nraizEntera x = aux (1,x)\n    where aux (a,b) | d == x    = c\n                    | c == a    = c\n                    | d < x     = aux (c,b)\n                    | otherwise = aux (a,c)\n              where c = (a+b) `div` 2\n                    d = c^2\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El 2\u00ba problema de la ONEM (Olimp\u00edada Nacional Escolar de Matem\u00e1tica) de Mayo del 2020 dice Determinar si existen enteros positivos a, b y c, no necesariamente distintos, tales que y es un cuadrado perfecto. Definir la funci\u00f3n soluciones :: Integer -> Integer -> [(Integer,Integer,Integer)] tales que (soluciones k n) es la lista de las&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[4],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6092"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=6092"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6092\/revisions"}],"predecessor-version":[{"id":6138,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6092\/revisions\/6138"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6092"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6092"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6092"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}