Question 125
Let $x$, $y$ and $z$ be real numbers such that $x + y + z = xy + yz + zx$. Prove that
$$
\frac{x}{\sqrt{x^4 + x^2 + 1}} + \frac{y}{\sqrt{y^4 + y^2 + 1}} + \frac{z}{\sqrt{z^4 + z^2 + 1}} \geq \frac{-1}{\sqrt{3}}
$$
$$
\frac{x}{\sqrt{x^4 + x^2 + 1}} + \frac{y}{\sqrt{y^4 + y^2 + 1}} + \frac{z}{\sqrt{z^4 + z^2 + 1}} \geq \frac{-1}{\sqrt{3}}
$$