\u3000\u6574\u6570 \\(a , b , c\\) \u306b\u95a2\u3059\u308b\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u8003\u3048\u308b.\r\n\\[\r\n\\displaystyle\\int_{a}^{c} ( x^2 +bx ) \\, dx = \\displaystyle\\int_{b}^{c} ( x^2 +ax ) \\, dx \\quad \\cdots ( \\text{\uff0a} )\r\n\\]\r\n
(1)<\/strong>\u3000\u6574\u6570 \\(a , b, c\\) \u304c (\uff0a) \u304a\u3088\u3073 \\(a \\neq b\\) \u3092\u307f\u305f\u3059\u3068\u304d,\r\n\\(c\\) \u306f \\(3\\) \u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(c = 3600\\) \u306e\u3068\u304d, (\uff0a) \u304a\u3088\u3073 \\(a \\lt b\\) \u3092\u307f\u305f\u3059\u6574\u6570\u306e\u7d44 \\(( a , b)\\) \u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\n& \\displaystyle\\int _ {a}^{c} ( x^2 +bx ) \\, dx = \\left[ \\dfrac{x^3}{3} +\\dfrac{b x^2}{2} \\right] _ {a}^{c} \\\\\r\n& \\qquad = \\dfrac{c^3}{3} +\\dfrac{b c^2}{2} -\\dfrac{a^3}{3} -\\dfrac{a^2 b}{2} \\ , \\\\\r\n& \\displaystyle\\int _ {b}^{c} ( x^2 +ax ) \\, dx = \\left[ \\dfrac{x^3}{3} +\\dfrac{a x^2}{2} \\right] _ {b}^{c} \\\\\r\n& \\qquad = \\dfrac{c^3}{3} +\\dfrac{a c^2}{2} -\\dfrac{b^3}{3} -\\dfrac{a b^2}{2}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, (\uff0a) \u3088\u308a\r\n\\[\\begin{align}\r\n3b c^2 -2 a^3 -3a^2 b = 3a c^2 -2 b^3 & -3a b^2 \\\\\r\n(b-a) \\left\\{ 3c^2 +2 ( a^2 +ab +b^2 ) +3ab \\right\\} & = 0 \\\\\r\n3 c^2 +2 a^2 +5ab +2 b^2 & = 0 \\quad ( \\ \\text{\u2235} \\ a \\neq b \\ ) \\\\\r\n\\text{\u2234} \\quad ( 2a +b ) ( b +2a ) & = -3c^2 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u53f3\u8fba\u306f \\(3\\) \u306e\u500d\u6570\u306a\u306e\u3067, \u5de6\u8fba\u3082 \\(3\\) \u306e\u500d\u6570\u3067\u3042\u308b. (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u7d4c\u904e\u304b\u3089 \\(2a+b = 3m \\ , \\ a+2b = 3n \\ ( m , n \\text{\u306f\u6574\u6570} )\\) \u3068\u304a\u304f\u3053\u3068\u304c\u3067\u304d\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n[1] \u306f \\(a . b\\) \u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, \\(2a +b = 3m \\ ( m \\text{\u306f\u6574\u6570})\\) \u3068\u8003\u3048\u3066\u3088\u3044.
\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\na +2b = 3 (a+b) -3m = 3( a +b -m )\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306e\u5de6\u8fba\u306f \\(9\\) \u306e\u500d\u6570\u3067\u3042\u308a, \u3088\u3063\u3066 \\(c\\) \u306f \\(3\\) \u306e\u500d\u6570\u3067\u3042\u308b.<\/p>\r\n
\r\n\u3053\u308c\u3092\u3068\u304f\u3068\r\n\\[\r\na = 2m -n \\ , \\ b = 2n -m \\quad ... [2]\r\n\\]\r\n\u306a\u306e\u3067, \\(a , b\\) \u306f\u3068\u3082\u306b\u6574\u6570\u3068\u306a\u308b.
\r\n\u307e\u305f, \\(-3 c^2 \\lt 0\\) \u306a\u306e\u3067 \\(m\\) \u3068\\(n\\) \u306f\u7570\u7b26\u53f7\u3067, \\(m \\lt 0 \\lt n\\) \u3068\u3059\u308c\u3070, [2] \u3088\u308a \\(a \\lt 0 \\lt b\\) \u3068\u306a\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u500b\u6570\u306f, \\(3c^2\\) \u3092 \\(2\\) \u3064\u306e\u56e0\u6570 \\(-3m\\) \u3068 \\(3n\\) \u306b\u5206\u3051\u308b\u65b9\u6cd5\u306b\u7b49\u3057\u3044.
\r\n\\(c = 3600 = 2^4 \\cdot 3^2 \\cdot 5^2\\) \u306e\u3068\u304d, \\(3c^2 = 2^8 \\cdot 3^5 \\cdot 5^4\\) \u3067\u3042\u308a,
\r\n\u7d20\u56e0\u6570 \\(2 , 3, 5\\) \u306e\u5206\u3051\u65b9\u304c\u305d\u308c\u305e\u308c \\(9 , 4 , 5\\) \u901a\u308a\u305a\u3064\u3042\u308b\u306e\u3067, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\r\n9 \\cdot 4 \\cdot 5 = \\underline{180}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u3000\u6574\u6570 \\(a , b , c\\) \u306b\u95a2\u3059\u308b\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u8003\u3048\u308b. \\[ \\displaystyle\\int_{a}^{c} ( x^2 +bx ) \\, dx = \\displaystyle\\int_{b}^{c … \u7d9a\u304d\u3092\u8aad\u3080