Inequality with the constraint $xy=x+y$ I have to prove that if $x$ and $y$ are two positive numbers such that $xy=x+y$, the following inequality
$$\frac{x}{y^2+4}+\frac{y}{x^2+4}\geq \frac{1}{2}$$
holds. Any help is appreciated, thank you in advance.

Since $\frac{(x+y)^2}{4}\geq xy=x+y$, we obtain $x+y\geq4$.
Thus, by C-S $\frac{x}{y^2+4}+\frac{y}{x^2+4}\geq\frac{(x+y)^2}{xy^2+x^2y+4(x+y)}=\frac{x+y}{xy+4}=\frac{1}{1+\frac{4}{x+y}}\geq\frac{1}{2}$.
