Question 37
Let $x$, $y$ and $z$ be positive real numbers such that $x + y + z = 18xyz$. Prove the inequality
$$
\frac{x}{\sqrt{x^2 + 2yz + 1}} + \frac{y}{\sqrt{y^2 + 2xz + 1}} + \frac{z}{\sqrt{z^2 + 2xy + 1}} \ge 1.
$$
$$
\frac{x}{\sqrt{x^2 + 2yz + 1}} + \frac{y}{\sqrt{y^2 + 2xz + 1}} + \frac{z}{\sqrt{z^2 + 2xy + 1}} \ge 1.
$$