Question 25
Show that for all positive real numbers $x$, $y$, $z$ satisfying $x^2 + y^2 + z^2 \le x + y + z$ the following inequality is held:
$$
\frac{x^2+3}{x^3+1} + \frac{y^2+3}{y^3+1} + \frac{z^2+3}{z^3+1} \ge 6
$$
$$
\frac{x^2+3}{x^3+1} + \frac{y^2+3}{y^3+1} + \frac{z^2+3}{z^3+1} \ge 6
$$