一道高一的数学题已知X+Y+Z=1/X+1/Y+1/Z=1,求证xyz中至少有一个是1

Question

一道高一的数学题
已知X+Y+Z=1/X+1/Y+1/Z=1,求证xyz中至少有一个是1

Answer 1

x+y+z=1/x+1/y+1/z=1
(x-1)(y-1)(z-1)
=(xyz)+x+y+z-xy-yz-zx-1
=(xyz)-xy-yz-zx
=(xyz)[1-1/x-1/y-1/z]
=(xyz)*0
=0
所以至少有一个是1
否则(x-1)(y-1)(z-1)不＝0