\\(a , b , c\\) \u3092\u76f8\u7570\u306a\u308b\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u6b21\u306e \\(2\\) \u6570\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088.\r\n\\[\r\na^3+b^3 , \\ a^2b+b^2a\r\n\\]<\/li>\r\n (2)<\/strong>\u3000\u6b21\u306e \\(4\\) \u6570\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088.\r\n\\[\\begin{align}\r\n& ( a+b+c ) ( a^2+b^2+c^2 ) , \\ ( a+b+c ) ( ab+bc+ca ) , \\\\\r\n& 3 ( a^3+b^3+c^3 ) , \\ 9abc\r\n\\end{align}\\]<\/li>\r\n (3)<\/strong>\u3000\\(x , y , z\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d\r\n\\[\r\n\\dfrac{y+z}{x} +\\dfrac{z+x}{y} +\\dfrac{x+y}{z}\r\n\\]\r\n\u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\na^3+b^3 & -( a^2b+b^2a ) \\\\\r\n& = a^2( a-b ) -b^2( a-b ) \\\\\r\n& = ( a-b )^2( a+b ) \\gt 0 \\quad ( \\ \\text{\u2235} \\ a-b \\neq 0 , \\ a+b \\gt 0 )\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\underline{a^3+b^3 \\gt a^2b+b^2a}\n\\]\r\n (2)<\/strong><\/p>\r\n \\(A = ( a+b+c ) ( a^2+b^2+c^2 )\\) , \\(B = ( a+b+c ) ( ab+bc+ca )\\) , \\(C = 3 ( a^3+b^3+c^3 )\\) , \\(D = 9abc\\) \u3068\u304a\u304f. (3)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\dfrac{y+z}{x} & + \\dfrac{z+x}{y} + \\dfrac{x+y}{z} \\\\\r\n& = \\dfrac{yz( y+z ) +zx( z+x ) + xy( x+y )}{xyz} \\\\\r\n& = \\dfrac{( x+y+z )( xy+yz+zx )}{xyz} -3 \\quad ... [4]\n\\end{align}\\]\r\n (2)<\/strong> \u306b\u304a\u3051\u308b [3] \u3092\u5c0e\u304f\u904e\u7a0b\u3092\u7528\u3044\u308c\u3070, \\((x-y)^2, (y-z)^2, (z-x)^2 \\geqq 0 , x, y, z \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n( x+y+z )( xy+yz+zx ) & \\geqq xyz \\\\\r\n\\dfrac{( x+y+z )( xy+yz+zx )}{xyz} -3 & \\geqq 6 \\\\\r\n\\underline{\\dfrac{y+z}{x} + \\dfrac{z+x}{y} + \\dfrac{x+y}{z} \\geqq 6} & \\quad ( \\ \\text{\u2235} \\ [4] \\ )\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(x=y=z\\) \u306e\u3068\u304d.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(a , b , c\\) \u3092\u76f8\u7570\u306a\u308b\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6b21\u306e \\(2\\) \u6570\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088. \\[ a^3+b^3 , \\ a^2b+b^2a \\] (2)\u3000\u6b21\u306e \\(4\\) \u6570\u306e\u5927 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(C\\) \u3068 \\(A\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nC -A & = 2 \\left( a^3+b^3+c^3 \\right) -a^2b -a^2c -b^2a -b^2c -c^2a -c^2b \\\\\r\n& = \\left( a^3 +b^3 -a^2b -b^2a \\right) +\\left( b^3 +c^3 -b^2c -c^2b \\right) \\\\\r\n& \\qquad +\\left( c^3 +a^3 -c^2a -a^2c \\right) \\gt 0 \\quad ( \\ \\text{\u2235} \\ \\text{(1)\u306e\u7d50\u679c} ) \\\\\r\n& \\text{\u2234} \\quad C \\gt A \\quad ... [1]\n\\end{align}\\]\r\n\\(A\\) \u3068 \\(B\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nA -B & = ( a+b+c ) \\left( a^2+b^2+c^2 -ab -bc -ca \\right) \\\\\r\n& = \\dfrac{1}{2} ( a+b+c ) \\left\\{ ( a-b )^2 +( b-c )^2 +( c-a )^2 \\right\\} \\\\\r\n& \\gt 0 \\quad ( \\ \\text{\u2235} \\ a-b, b-c, c-a \\neq 0 , \\ a+b+c \\gt 0\\ ) \\\\\r\n& \\text{\u2234} \\quad A \\gt B \\quad ... [2]\n\\end{align}\\]\r\n\\(B\\) \u3068 \\(D\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nB -D & = ab( a+b ) +bc( b+c ) +ca( c+a ) -6abc \\\\\r\n& = a( b^2 +c-2 -2bc ) +b( c^2 +a^2 -2ca ) + c( a^2 +b^2 -2ab ) \\\\\r\n& = a( b-c )^2 +b( c-a )^2 +c( a-b )^2 \\\\\r\n& \\gt 0 \\quad ( \\ \\text{\u2235} \\ a-b, b-c, c-a \\neq 0 , \\ a, b, c \\gt 0\\ ) \\\\\r\n& \\text{\u2234} \\quad B \\gt D \\quad ... [3]\n\\end{align}\\]\r\n[1] \uff5e [3] \u3088\u308a\r\n\\[\\begin{align}\r\n& \\underline{3 ( a^3+b^3+c^3 ) \\gt ( a+b+c ) ( a^2+b^2+c^2 )} \\\\\r\n& \\qquad \\underline{\\gt ( a+b+c ) ( ab+bc+ca ) \\gt 9abc}\n\\end{align}\\]\r\n