KöMaL 2011 B4355

Prove that if the product of the positive numbers $x$, $y$ and $z$ is 1, then $\frac{z^3 + y^3}{x^2+xy+y^2} + \frac{x^3 + z^3}{y^2+yz+z^2} + \frac{y^3 + x^3}{z^2+zx+x^2} \ge 2.$ (Based on the idea of J. Szoldatics, Dunakeszi)