Question

已知X,Y,Z为不相等的实数,且X+1/Y=Y+1/Z=Z+1/X,求X²Y²Z²

Answer 1

因为x+1/y=y+1/z所以x-y=1/z-1/y即x-y=(y-z)/yz
同理y-z=(z-x)/xz,z-x=(x-y)/xy
所以x-y=(y-z)/yz=(z-x)/xyz^2=(x-y)/x^2y^2z^2
又x不等于y不等于z,即x-y不为0
所以x^2y^2z^2=1

Answer 2

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