作者Honor1984 (奈何上天造化弄人？)
看板Math
标题Re: [中学] 9^x+9^y+9^z=3^x+y + 3^y+z + 3^z+x
时间Wed Sep 27 00:19:45 2023
※ 引述《Dirichlet (微风轻吹)》之铭言：
: 设 x,y,z 为三角形的三边长,且满足
: x^9 + y^9 + z^9 = 3^(x+y) + 3^(y+z) + 3^(z+x)
: 问此三角形是何种三角形？
文章内题目忘了改
9^x + 9^y + 9^z = 3^(x+y) + 3^(y+z) + 3^(z+x)
令A = 3^x，B = 3^y，C = 3^z
2A^2 + 2B^2 + 2C^2 = 2AB + 2BC + 2CA
=> (A - B)^2 + (B - C)^2 + (C - A)^2 = 0
=> A = B = C
=> x = y = z
=> 正三角形
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