\u81ea\u7136\u6570 \\(m , n\\) \u306b\u5bfe\u3057\u3066 \\(f(m,n)\\) \u3092\r\n\\[\r\nf(m,n) = \\dfrac{1}{2} \\{ (m+n-1)^2 +(m-n+1) \\}\r\n\\]\r\n\u3067\u5b9a\u3081\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(f(m,n) =100\\) \u3092\u307f\u305f\u3059 \\(m\\) , \\(n\\) \u3092 \\(1\\) \u7d44\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(a , b , c , d\\) \u306f\u6574\u6570\u3067, \u7b49\u5f0f \\(a^2+b = c^2+d\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \u4e0d\u7b49\u5f0f \\(-a \\lt b \\leqq a , \\ -c \\lt d \\leqq c\\) \u304c\u6210\u308a\u7acb\u3064\u306a\u3089\u3070, \\(a = c , \\ b = d\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\u4efb\u610f\u306e\u81ea\u7136\u6570 \\(k\\) \u306b\u5bfe\u3057, \\(f(m,n) =k\\) \u3092\u307f\u305f\u3059 \\(m , n\\) \u304c\u305f\u3060 \\(1\\) \u7d44\u3060\u3051\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\r\n(m+n-1)^2 +(m-n+1) =200\r\n\\]\r\n\u3053\u3053\u3067, \\(m+n-1 = 14\\) ... [1] \u3068\u304a\u304f\u3068\r\n\\[\r\nm-n+1 = 4 \\quad ... [2]\r\n\\]\r\n[1] [2] \u3092\u89e3\u304f\u3068\r\n\\[\r\n(m,n) = \\underline{( 9, 6 )}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \u4e0e\u3048\u3089\u308c\u305f\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na^2 -c^2 & = d-b \\\\\r\n\\text{\u2234} \\quad (a+c)(a-c) & = d-b \\quad ... [3]\r\n\\end{align}\\]\r\n\u4e00\u65b9, \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3088\u308a \\(a \\gt 0 , \\ c \\gt 0\\) \u3067\u3042\u308a\r\n\\[\r\n-(a+c) \\lt d-b \\lt a+c\r\n\\]\r\n[3] \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n-(a+c) & \\lt (a+c)(a-c) \\lt a+c \\\\\r\n\\text{\u2234} \\quad -1 & \\lt a-c \\lt 1\r\n\\end{align}\\]\r\n\\(a-c\\) \u306f\u6574\u6570\u306a\u306e\u3067\r\n\\[\\begin{align}\r\na-c & = 0 \\\\\r\n\\text{\u2234} \\quad a & = c\r\n\\end{align}\\]\r\n[3] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nd-b & = 0 \\\\\r\n\\text{\u2234} \\quad b & = d\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (3)<\/strong><\/p>\r\n \\[\\begin{align}\r\n(m+n-1)^2 +(m-n+1) & = 2k \\\\\r\n(m+n-1)^2 +(m+n-1) & = 2(k+n-1) \\\\\r\n\\text{\u2234} \\quad \\dfrac{(m+n-1)(m+n)}{2} & = k+n-1\r\n\\end{align}\\]\r\n\\(S _ {\\ell} = 0 +1+2+ \\cdots +\\ell\\) \uff08 \\(\\ell\\) \u306f\u975e\u8ca0\u6574\u6570\uff09\u3068\u304a\u304f\u3068\r\n\\[\r\nS _ {m+n-1} = k+n-1 \\quad ... [4]\r\n\\]\r\n\\(S _ {\\ell}\\) \u306f\u5358\u8abf\u5897\u52a0\u6570\u5217\u3060\u304b\u3089, \\(S _ {\\ell-1} \\lt k \\leqq S _ {\\ell}\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(\\ell\\) \u304c\u305f\u3060 \\(1\\) \u3064\u5b58\u5728\u3059\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(m+n-1 =\\ell\\) ... [5] \u3068\u304a\u3051\u3070, [4] \u3088\u308a\r\n\\[\\begin{align}\r\nS _ {\\ell} & = k +\\ell -m \\\\\r\n\\text{\u2234} \\quad m & = k +\\ell -S _ {\\ell}\r\n\\end{align}\\]\r\n[5] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\nn = S _ {\\ell}-k +1\r\n\\]\r\n\u3088\u3063\u3066, \\(k\\) \u306b\u3088\u3063\u3066\u4e00\u610f\u306b\u6c7a\u307e\u308b\u81ea\u7136\u6570 \\(\\ell\\) \u3082\u7528\u3044\u3066, \\((m,n)\\) \u306e\u7d44\u306f\u305f\u3060 \\(1\\) \u7d44\r\n\\[\r\n(m,n) = \\left( k +\\ell -S _ {\\ell} , S _ {\\ell}-k +1 \\right)\r\n\\]\r\n\u3068\u5b9a\u307e\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u81ea\u7136\u6570 \\(m , n\\) \u306b\u5bfe\u3057\u3066 \\(f(m,n)\\) \u3092 \\[ f(m,n) = \\dfrac{1}{2} \\{ (m+n-1)^2 +(m-n+1) \\} \\] \u3067\u5b9a\u3081\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(f(m … \u7d9a\u304d\u3092\u8aad\u3080