knowing: x+1/y=y+1/z=z+1/x, and x≠y≠z; to testfy x*x*y*y*z*z=1?

Answer 1

Taking the equations in pairs and simpliyfing, we have:

yz(x-y) = (y-z)
zx(y-z) = (z-x)
xy(z-x) = (x-y)

Multiplying all of them together gets us:

yzzxxy(x-y)(y-z)(z-x) = (y-z)(z-x)(x-y)

or

yzzxxy = 1, unless (x-y), (y-z), or (z-x) = 0