\\(x , y\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(\\dfrac{2}{x} +\\dfrac{1}{y}= \\dfrac{1}{4}\\) \u3092\u307f\u305f\u3059\u7d44 \\((x, y)\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(p\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u7d20\u6570\u3068\u3059\u308b. \\(\\dfrac{2}{x} +\\dfrac{1}{y}= \\dfrac{1}{p}\\) \u3092\u307f\u305f\u3059\u7d44 \\((x, y)\\) \u306e\u3046\u3061, \\(2x+3y\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\((x, y)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4e0e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\n8y+4x & = xy \\\\\r\n\\text{\u2234} \\quad (x-8)(y-4) & =32\n\\end{align}\\]\r\n\\(x \\geqq 1 , \\ y \\geqq 1\\) \u3088\u308a, \\(x-8 \\geqq -7 , \\ y-4 \\geqq -3\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n( x-8 , y-4 ) & = ( 1, 32 ) , ( 2, 16 ) , ( 4, 8 ) , ( 8, 4 ) , ( 16, 2 ) , ( 32, 1 ) \\\\\r\n\\text{\u2234} \\quad ( x, y ) & = \\underline{( 9 , 36 ) , ( 10 , 20 ) , ( 12 , 12 ) , ( 16 , 8 ) , ( 24 , 6 ) , ( 40 , 5 )}\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \u4e0e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\n2py+px & =xy \\\\\r\n\\text{\u2234} \\quad (x-2p)(y-p) & = 2p^2\n\\end{align}\\]\r\n\\(x \\geqq 1 , \\ y \\geqq 1\\) \u3088\u308a, \\(x-2p \\geqq 1-2p , \\ y-p \\geqq 1-p\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n( x-2p , y-p ) & = ( 1 , 2p^2 ) , ( 2 , p^2 ) , ( p , 2p ) , ( 2p , p ) , ( p^2 , 2 ) , ( 2p^2 , 1 ) \\\\\r\n\\text{\u2234} \\quad ( x , y ) & = \\left( 2p+1 , p(2p+1) \\right) , \\left( 2(p+1) , p(p+1) \\right) , \\\\\r\n& \\qquad \\left( 3p , 3p \\right) , \\left( 4p , 2p \\right) , \\left( p(p+2) , p+2 \\right) , \\left( 2p(p+1) , p+1 \\right)\n\\end{align}\\]\r\n\u3053\u306e\u3046\u3061, \\(2x+3y\\) \u3092\u6700\u5c0f\u306b\u3059\u308b\u3082\u306e\u3092\u63a2\u3059.\r\n\\[\r\n2 \\cdot 4p +3 \\cdot 2p =14p\n\\]\r\n\u3053\u308c\u306b\u5bfe\u3057\u3066, \\(p \\geqq 3\\) \u3092\u7528\u3044\u3066, \u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n& 2 \\cdot 3p +3 \\cdot 3p -14p =p \\gt 0 , \\\\\r\n& 2 (2p+1) +3 p(2p+1) -14p = 6p^2 -7p +2 \\\\\r\n& \\qquad \\geqq 11p+2 \\gt 0 , \\\\\r\n& 2 \\cdot 2(p+1) +3 p(p+1) -14p = 3p^2 -7p +2 \\\\\r\n& \\qquad \\geqq 2p+2 \\gt 0 , \\\\\r\n& 2 \\cdot p(p+2) +3 (p+2) -14p = 2p^2 -7p +6 \\\\\r\n& = 2\\left( p -\\dfrac{7}{4} \\right)^2 -\\dfrac{1}{8} \\gt 0 , \\\\\r\n& 2 \\cdot 2p(p+1) +3 (p+1) -14p = 4p^2 -7p +3 \\\\\r\n& \\qquad \\geqq 5p+3 \\gt 0\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u6700\u5c0f\u3068\u306a\u308b\u7d44\u306f\r\n\\[\r\n\\underline{( 4p , 2p )}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(x , y\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. (1)\u3000\\(\\dfrac{2}{x} +\\dfrac{1}{y}= \\dfrac{1}{4}\\) \u3092\u307f\u305f\u3059\u7d44 \\((x, y)\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088. (2)\u3000\\(p\\) \u3092 \\(3 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n