Question 54
Prove that the following inequality is satisfied for any positive real numbers $x$, $y$, $z$:
$$
\frac{x^3}{y^2 + z^2} + \frac{y^3}{x^2 + z^2} + \frac{z^3}{x^2 + y^2} \ge \frac{x + y + z}{2}
$$
$$
\frac{x^3}{y^2 + z^2} + \frac{y^3}{x^2 + z^2} + \frac{z^3}{x^2 + y^2} \ge \frac{x + y + z}{2}
$$