已知x、y、z为三个不相等的实数,且x+1/y=y+1/z=z+1/x,则x^2*y^2*z^2=?回答请尽量详细,

Question

已知x、y、z为三个不相等的实数,且x+1/y=y+1/z=z+1/x,则x^2*y^2*z^2=?
回答请尽量详细,

Answer 1

由x+1/y=y+1/z得x-y=(y-z)/yz (1),再由x+1/y=z+1/x得x-z=1/x-1/y=(y-x)/xy,再将(1)代入得xyyz=(z-y)/(x-z) (2)
同理,xxyz=(x-y)/(y-z) (3),xyzz=(z-x)/(x-y) (4)
(2)(3)(4)相乘得xyz=1
x^2*y^2*z^2=1