设x，y，z为互不相等的非零实数，且x+1y＝y+1z＝z+1x．求证：x2y2z2=1．

Question

设x，y，z为互不相等的非零实数，且x+(1)/(y)＝y+(1)/(z)＝z+(1)/(x)．求证：x²y²z²=1．

Answer 1

证明：由已知x+(1)/(y)＝y+(1)/(z)＝z+(1)/(x)得出：
∵x+(1)/(y)=y+(1)/(z)，
∴x-y=(1)/(z)−(1)/(y)，
x-y=(y−z)/(yz)，
∴yz=(y−z)/(x−y)，①
同理得出
zx=(z−x)/(y−z)，②
xy=(x−y)/(z−x)．③
①×②×③得x²y²z²=1．