BALKAN 2011 Problem 2

Given real numbers $x, y, z$ such that $x+y+z=0$, show that

$$\frac{x(x+2)}{2 x^{2}+1}+\frac{y(y+2)}{2 y^{2}+1}+\frac{z(z+2)}{2 z^{2}+1} \geq 0 .$$

When does equality hold?