Question 47
Let $x$, $y$, $z$ be positive real numbers such that $x + y + z = 9xyz$. Prove that:
$$
\frac{x}{\sqrt{x^2 + 2yz + 2}} + \frac{y}{\sqrt{y^2 + 2zx + 2}} + \frac{z}{\sqrt{z^2 + 2xy + 2}} \geq 1.
$$
When does the equality hold?
$$
\frac{x}{\sqrt{x^2 + 2yz + 2}} + \frac{y}{\sqrt{y^2 + 2zx + 2}} + \frac{z}{\sqrt{z^2 + 2xy + 2}} \geq 1.
$$
When does the equality hold?