TUYMAADA 2013 S4

Prove that if $x$ , $y$ , $z$ are positive real numbers and $xyz = 1$ then $$\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.$$

A. Golovanov