已知x.y.z是三个不相等的实数,且x+1/y=y+1/z=z+1/x,求x^2y^2z^2=1

Question

已知x.y.z是三个不相等的实数,且x+1/y=y+1/z=z+1/x,求x^2y^2z^2=1

Answer 1

因为 x+1/y=y+1/z
所以 x-y=1/z-1/y=(y-x)/zy
同理 y-z=(z-x)/xz
z-x=(x-y)/yx=(y-x)/xzy^2
=(z-x)/x^2y^2z^2
两边都约去z-x得到x^2y^2z^2=1