- 高中数学
- 设x,y,z为互不相等实数,且x+1/y=y+1/z=z+1/x.
求证:x^2*y^2*z^2=1
- 证明 x,y,z为互不相等实数,
由x+1/y=y+1/z,可得 x-y=1/z-1/y
即 (y-z)/(x-y)=yz (1)
同理可得:
x+1/y=z+1/x ==>
(x-y)/(z-x)=xy; (2)
y+1/z=z+1/x ==>
(z-x)/(y-z)=zx (3)
(1)*(2)*(3)得:x^2y^2z^2=1.
证明:由x+1/y=y+1/z,得x-y=1/z-1/y
即x-y=(y-z)/yz
同理y-z=(z-x)/xz,z-x=(x-y)/xy
故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