Math contest proof problem fractions Could someone help me with this?
Let $x, y, z$ be positive integers with greatest common divisor $1$. If $\frac 1 x +\frac 1 y=\frac 1 z$, then show that $\sqrt{x + y}$ is an integer.

Note: we can replace the condition $$\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$$ with the following weaker one:

For every prime $p$, $$\nu_p(x+y)=\nu_p(x)+\nu_p(y)-\nu_p(z)=S_p\geq 0$$

First, suppose that $\nu_p(z)=0$. Then $\nu_p(x+y)=\nu_p(x)+\nu_p(y)$, so $\nu_p(x)=\nu_p(y)$ and $S_p$ is even.
Otherwise, suppose wlog that $\nu_p(x)=0$. If $\nu_p(y)=0$, then $S_p=0$, so suppose $\nu_p(y)\geq 1$. Then $\nu_p(x+y)=0\implies S_p=0$ again.
Thus in all cases, $S_p$ is even. It follows that $x+y$ is a perfect square.
