推 Starvilo : =(x+y+z)0.5[(x-y)^2+(y-z)^2+(z-x)^2)=0 x=y=z 03/09 11:50
※ 引述《sam206 ()》之銘言：
: 2)設x,y,z均為正整數，滿足 x^3+y^3+z^3=3xyz，且x^2=2(y+z)，求x,y,z
(x^2 + y^2 + z^2)(x + y + z)
= (x^3 + y^3 + z^3) + ( x^2(y+z) + y^2(x+z) + z^2(x+y) )
= 3xyz + ( x^2(y+z) + y^2(x+z) + z^2(x+y) )
= (xy + yz + xz)(x + y + z)
因為 x, y, z 均為正整數
=> xy + yz + xz = x^2 + y^2 + z^2
x^2 - (y + z)x + (y^2 + z^2 - yz) = 0
x 判別式 D = (y + z)^2 - 4(y^2 + z^2 - yz)
= -3(y^2 - 2yz + z^2)
= -3(y - z)^2
>= 0
=> y = z
同理， x = y = z
又 x^2 = 2(x+x) => x = y = z = 4
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