Question 111
Let $x, y, z \gt 0$. Show that
$$
\frac{x^{3}}{z^{3}+x^{2} y} + \frac{y^{3}}{x^{3}+y^{2} z} + \frac{z^{3}}{y^{3}+z^{2} x} \geq \frac{3}{2}
$$
$$
\frac{x^{3}}{z^{3}+x^{2} y} + \frac{y^{3}}{x^{3}+y^{2} z} + \frac{z^{3}}{y^{3}+z^{2} x} \geq \frac{3}{2}
$$