BALTIC-WAY 2013 Problem 4

Prove that the following inequality holds for all positive real numbers $x, y, z$ :

$$\frac{x^{3}}{y^{2}+z^{2}}+\frac{y^{3}}{z^{2}+x^{2}}+\frac{z^{3}}{x^{2}+y^{2}} \geq \frac{x+y+z}{2}$$