if x+1/y=y+1/z=z+1/x,then prove x=y=z

Solution:

We can prove this by contradiction.

Suppose x = y then x = z. So you can assume they are all different, say x > y > z.

If, they are all of the same sign, say z > 0.

Then y > z and 1/z > 1/y > 1/x imply y + 1/z > z + 1/x.

If the signs are split, say x > y > 0 > z. In this case x + 1/y > y + 1/z.

Contradiction again. That's it.