Question 170
Let $x$, $y$, $z$ be positive real numbers such that $x + y + z \ge 6$. Find the smallest value of the expression
$$
x^2 + y^2 + z^2 + \frac{x}{y^2 + z + 1} + \frac{y}{z^2 + x + 1} + \frac{z}{x^2 + y + 1}.
$$
$$
x^2 + y^2 + z^2 + \frac{x}{y^2 + z + 1} + \frac{y}{z^2 + x + 1} + \frac{z}{x^2 + y + 1}.
$$