\u6b21\u306e\u547d\u984cP\u3092\u8a3c\u660e\u3057\u305f\u3044.<\/p>\r\n
\u547d\u984cP\uff1a\u6b21\u306e\u6761\u4ef6 (a)<\/strong> , (b)<\/strong> \u3092\u3068\u3082\u306b\u307f\u305f\u3059\u81ea\u7136\u6570\uff08 \\(1\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09 \\(A\\) \u304c\u5b58\u5728\u3059\u308b.<\/p>\r\n (a)<\/strong>\u3000\\(A\\) \u306f\u9023\u7d9a\u3059\u308b \\(3\\) \u3064\u306e\u81ea\u7136\u6570\u306e\u7a4d\u3067\u3042\u308b.<\/p><\/li>\r\n (b)<\/strong>\u3000\\(A\\) \u3092 \\(10\\) \u9032\u6cd5\u3067\u8868\u3057\u305f\u3068\u304d, \\(1\\) \u304c\u9023\u7d9a\u3057\u3066 \\(99\\) \u56de\u4ee5\u4e0a\u73fe\u308c\u308b\u3068\u3053\u308d\u304c\u3042\u308b.<\/p><\/li>\r\n<\/ol><\/li>\r\n<\/ol>\r\n \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n (1)<\/strong>\u3000\\(y\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u4e0d\u7b49\u5f0f\r\n\\[\r\nx^3+3yx^2 \\lt (x+y-1)(x+y)(x+y+1) \\lt x^3 +(3y+1)x^2\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u6b63\u306e\u5b9f\u6570 \\(x\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u547d\u984cP\u3092\u8a3c\u660e\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092 [\uff03] \u3068\u304a\u304f. (2)<\/strong><\/p>\r\n \u300c \\(1\\) \u300d\u304c \\(99\\) \u500b, \u3059\u306a\u308f\u3061\u300c \\(111\\) \u300d\u304c \\(33\\) \u500b\u4e26\u3093\u3067\u3067\u304d\u305f\u81ea\u7136\u6570\u3092 \\(3y\\) \u3068\u304a\u304f.\r\n
\r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n[\uff03] \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n& x^3 +3yx^2 \\lt (x+y)^3 -(x+y) \\lt x^3 +(3y+1)x^2 \\\\\r\n& \\qquad 0 \\lt 3y^2x+y^3-x-y \\lt x^2 \\\\\r\n& \\text{\u2234} \\quad \\left\\{ \\begin{array}{ll} (3y^2-1)x+y(y^2-1) \\gt 0 & \\ ... \\text{[A]} \\\\ x^2-(3y^2-1)x-y(y^2-1) \\gt 0 & \\ ...\\text{[B]} \\end{array} \\right.\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [A] \u3068 [B] \u304c\u6210\u308a\u7acb\u3064\u305f\u3081\u306e \\(y\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n\r\n
\r\n\\(y\\) \u306f\u81ea\u7136\u6570\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n3y^2-1 \\geqq 2 & \\gt 0 \\quad ...[1] \\\\\r\ny^3-y = y(y^2-1) & \\geqq 0 \\quad ...[2]\r\n\\end{align}\\]\r\n\u3055\u3089\u306b \\(x \\gt 0\\) \u306a\u306e\u3067, [A]\u306f \\(y\\) \u306b\u3088\u3089\u305a\u5e38\u306b\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n
\r\n\u5224\u5225\u5f0f\u3092 \\(D\\) \u3068\u3059\u308b\u3068\r\n\\[\r\nD = (3y^2-1)^2+4y(y^2-1) \\quad ... [3]\r\n\\]\r\n[1] [2] \u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\nD \\gt 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [B] \u3092\u3068\u304f\u3068\r\n\\[\r\nx \\lt \\dfrac{3y^2-1 -\\sqrt{D}}{2} , \\dfrac{3y^2-1 +\\sqrt{D}}{2} \\lt x\r\n\\]\r\n[3] \u3088\u308a \\(\\sqrt{D} \\gt 3y^2-1\\) \u3067\u3042\u308a, \u3055\u3089\u306b \\(x \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\nx \\gt \\dfrac{3y^2-1 +\\sqrt{D}}{2}\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(x\\) \u306e\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{x \\gt \\dfrac{3y^2-1 +\\sqrt{9y^4+4y^3-6y^2-4y+1}}{2}}\r\n\\]<\/li>\r\n<\/ol>\r\n
\r\n\u3059\u308b\u3068, \\(y\\) \u306f\u300c \\(037\\) \u300d\u304c \\(33\\) \u500b\u4e26\u3093\u3067\u3067\u304d\u305f\u81ea\u7136\u6570\u3067\u3042\u308b.\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\ny = 37 \\cdot 10^{3 \\cdot 32} + \\cdots + 37 \\cdot 10^3 +37 \\quad ...[4]\r\n\\]\r\n\\(10^{97} \\lt y \\lt 10^{98}\\) \u3067\u3042\u308a, \\(\\sqrt{D} \\lt 2 (3y^2-1)\\) \u306a\u306e\u3067\r\n\\[\r\n\\dfrac{3y^2-1+\\sqrt{D}}{2} \\lt \\dfrac{3 ( 3y^2-1 )}{2} \\lt 10^{2 \\cdot 99}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(x = 10^{200}\\) ...[5] \u3068\u304a\u3051\u3070 \\(x\\) \u306f [\uff03] \u3092\u307f\u305f\u3059.\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nx^3+3yx^2 & = 10^{600} +3y \\cdot 10^{400} \\\\\r\n& = 1 \\underbrace{00 \\cdots 0} _ {501 \\text{\u500b}} \\underbrace{11 \\cdots 11} _ {99 \\text{\u500b}} \\underbrace{00 \\cdots 0} _ {200 \\text{\u500b}} \\\\\r\nx^3+(3y+1)x^2 & = 10^{600} +(3y+1) \\cdot 10^{400} \\\\\r\n& = 1 \\underbrace{00 \\cdots 0} _ {501 \\text{\u500b}} \\underbrace{11 \\cdots 1} _ {98 \\text{\u500b}} 2 \\underbrace{00 \\cdots 0} _ {200 \\text{\u500b}}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [\uff03] \u306e\u4e2d\u9805\u306f\u4e0b\u304b\u3089 \\(201\\) \u6841\u76ee\u304b\u3089 \\(299\\) \u6841\u76ee\u306b \\(1\\) \u304c \\(99\\) \u500b\u4e26\u3093\u3060\u6570\u3092\u8868\u3059.
\r\n\u3057\u305f\u304c\u3063\u3066, [4] [5] \u3068\u304a\u3044\u305f\u3068\u304d, \\(3\\) \u3064\u306e\u9023\u7d9a\u3059\u308b\u81ea\u7136\u6570\u306e\u7a4d \\(A = (x+y-1)(x+y)(x+y+1)\\) \u306f, \u6761\u4ef6 (b)<\/strong> \u3092\u6e80\u305f\u3059.
\r\n\u3088\u3063\u3066, \u547d\u984cP\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u547d\u984cP\u3092\u8a3c\u660e\u3057\u305f\u3044. \u547d\u984cP\uff1a\u6b21\u306e\u6761\u4ef6 (a) , (b) \u3092\u3068\u3082\u306b\u307f\u305f\u3059\u81ea\u7136\u6570\uff08 \\(1\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09 \\(A\\) \u304c\u5b58\u5728\u3059\u308b. (a)\u3000\\(A\\) \u306f\u9023\u7d9a\u3059\u308b \\(3\\) \u3064\u306e\u81ea\u7136\u6570\u306e\u7a4d\u3067\u3042\u308b. (b)\u3000\\( … \u7d9a\u304d\u3092\u8aad\u3080