{"id":414,"date":"2012-06-02T11:51:12","date_gmt":"2012-06-02T02:51:12","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=414"},"modified":"2021-11-04T09:10:36","modified_gmt":"2021-11-04T00:10:36","slug":"htb201201","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201201\/","title":{"rendered":"\u4e00\u6a4b\u59272012\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(1\\) \u3064\u306e\u89d2\u304c \\(120^{\\circ}\\) \u306e\u4e09\u89d2\u5f62\u304c\u3042\u308b.\r\n\u3053\u306e\u4e09\u89d2\u5f62\u306e \\(3\\) \u8fba\u306e\u9577\u3055 \\(x , y , z\\) \u306f \\(x \\lt y \\lt z\\) \u3092\u6e80\u305f\u3059\u6574\u6570\u3067\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(x+y-z = 2\\) \u3092\u6e80\u305f\u3059 \\(x , y , z\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x+y-z = 3\\) \u3092\u6e80\u305f\u3059 \\(x , y , z\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a , b\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \\(x+y-z = 2^a 3^b\\) \u3092\u6e80\u305f\u3059 \\(x , y , z\\) \u306e\u7d44\u306e\u500b\u6570\u3092 \\(a\\) \u3068 \\(b\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    \\(x+y-z = k\\) ... [1] \u3068\u304a\u304f\u3068, \\(z = x+y-k\\) .
    \r\n\u4f59\u5f26\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\nx^2 +y^2 -2xy \\cos 120^{\\circ} & = (x+y-k)^2 \\\\\r\n(x+y)^2 -xy & = (x+y)^2 -2k(x+y) +k^2 \\\\\r\n\\text{\u2234} \\quad (x-2k)(y-2k) & = 3k^2 \\quad ... [2]\r\n\\end{align}\\]\r\n

    (1)<\/strong><\/p>\r\n[1] \u306b\u304a\u3044\u3066, \\(k = 2\\) \u306e\u3068\u304d\u306a\u306e\u3067, [2] \u3088\u308a\r\n\\[\r\n(x-4)(y-4) = 12\r\n\\]\r\n\\(12 = 2^2 \\cdot 3\\) \u3067 \\(x \\lt y\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n(x-4 , y-4) & = (1,12) , (2,6) , (3,4) \\\\\r\n\\text{\u2234} \\quad (x,y) & = (5,16) , (6,10) , (7,8)\r\n\\end{align}\\]\r\n\u305d\u308c\u305e\u308c\u306b\u5bfe\u5fdc\u3059\u308b \\(z\\) \u306f\r\n\\[\r\nz = 19 , 14 , 13\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n(x,y,z) =\\underline{(5,16,19) , (6,10,14) , (7,8,13)}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n[1] \u306b\u304a\u3044\u3066, \\(k = 3\\) \u306e\u3068\u304d\u306a\u306e\u3067, [2] \u3088\u308a\r\n\\[\r\n(x-6)(y-6) = 27\r\n\\]\r\n\\(27 = 3^3\\) \u3067 \\(x \\lt y\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n(x-6 , y-6) & = (1,27) , (3,9) \\\\\r\n\\text{\u2234} \\quad (x,y) & = (7,33) , (9,15)\r\n\\end{align}\\]\r\n\u305d\u308c\u305e\u308c\u306b\u5bfe\u5fdc\u3059\u308b \\(z\\) \u306f\r\n\\[\r\nz = 37 , 21\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n(x,y,z) =\\underline{(7,33,37) , (6,10,21)}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n[1] \u306b\u304a\u3044\u3066, \\(k = 2^a 3^b\\) \u306e\u3068\u304d\u306a\u306e\u3067, [2] \u3088\u308a\r\n\\[\r\n(x -2^{a+1} 3^b)(y -2^{a+1} 3^b) = 2^{2a} 3^{2b+1} \\quad ... [3]\r\n\\]\r\n\\(a^{2a} 3^{2b+1}\\) \u306f\u5e73\u65b9\u6570\u3067\u306f\u306a\u304f, \u7d04\u6570\u3092 \\((2a+1)(2b+2)\\) \u500b\u3082\u3064\u306e\u3067, [3] \u3092\u307f\u305f\u3059 \\((x,y)\\) \u306e\u7d44\u306e\u6570\u306f\r\n\\[\r\n\\dfrac{1}{2} (2a+1)(2b+2) = (2a+1)(b+1)\r\n\\]\r\n\u3053\u3053\u3067, \\(x \\lt y \\lt 2k\\) \u306a\u306e\u3067\r\n\\[\r\nz = x+y-k \\gt 2k+2k-k = 3k\r\n\\]\r\n\u306a\u306e\u3067, [3] \u306e\u89e3\u3059\u3079\u3066\u306b\u3064\u3044\u3066, \\(z\\) \u306f \\(x \\lt y \\lt z\\) \u3092\u307f\u305f\u3059.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\r\n\\underline{(2a+1)(b+1)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u3064\u306e\u89d2\u304c \\(120^{\\circ}\\) \u306e\u4e09\u89d2\u5f62\u304c\u3042\u308b. \u3053\u306e\u4e09\u89d2\u5f62\u306e \\(3\\) \u8fba\u306e\u9577\u3055 \\(x , y , z\\) \u306f \\(x \\lt y \\lt z\\) \u3092\u6e80\u305f\u3059\u6574\u6570\u3067\u3042\u308b. (1)\u3000\\(x+y-z … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[67],"tags":[146,68],"class_list":["post-414","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2012","tag-hitotsubashi","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/414","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=414"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/414\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=414"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=414"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}