已知x+1y=y+1z=z+1x，其中x、y、z互不相等，求证：x2y2z2=1．

Question

已知x+(1)/(y)=y+(1)/(z)=z+(1)/(x)，其中x、y、z互不相等，求证：x²y²z²=1．

Answer 1

由x+(1)/(y)=y+(1)/(z)=z+(1)/(x)可得：x-y=(1)/(z)-(1)/(y)，zy=(y-z)/(x-y)，
同理，zx=(z-x)/(y-z)，xy=(y-x)/(x-z)，
∴x²y²z²=(y-z)/(x-y)×(x-z)/(y-z)×(y-x)/(x-z)=1．
故结论得证．