已知x.y.z是三个互不相等的实数,且x+1/y=y+1/z=z+1/x,求证xyz的绝对值=1

Question

已知x.y.z是三个互不相等的实数,且x+1/y=y+1/z=z+1/x,求证xyz的绝对值=1

Answer 1

x+1/y=y+1/z=z+1/x
有：
x-y=1/z-1/y
y-z=1/x-1/z
z-x=1/y-1/x
三式相乘
（x-y)(y-z)(z-x)=(1/z-1/y)(1/x-1/z)(1/y-1/x) 右边统分
（x-y)(y-z)(z-x)=（x-y)(y-z)(z-x)/(xyz)^2
(xyz)^2=1
|xyz|=1