I need to solve this problem and I don’t know how:
if $x + \frac 1y = y + \frac 1z = z + \frac 1x$, then $x=y=z$ or $x^2y^2z^2 = 1.$
I don’t need a full solution -- just a hint -- because I really don’t know what to do here. I tried to make a system like
$xy + 1 = ky ,$ $zy + 1 = zk ,$ $xz + 1 = xk .$
Then I tried to add and subtract different equations in different ways, but I don’t know if I am even going in the right direction, so I’d appreciate a hint. Thanks.
$$x+\frac{1}{y}=y+\frac{1}{z}$$ it's $$yz(x-y)=y-z.$$ Similarly, $$zx(y-z)=z-x$$ and $$xy(z-x)=x-y.$$ Now, if $x=y$ so $y=z$ and we obtain $x=y=z.$
Let $(x-y)(x-z)(y-z)\neq0.$
Thus, $$x^2y^2z^2(x-y)(y-z)(z-x)=(x-y)(y-z)(z-x)$$ or $$x^2y^2z^2=1.$$