Question 134
Let $x$, $y$, $z$ be positive real numbers such that $x + y + z = 1$. Prove that the following inequality holds:
$$
\frac{\sqrt{xyz}}{x^2 + y^2 + z^2 - x^3 - y^3 - z^3} \le \sqrt{\frac{xy}{(1-z)^2} + \frac{yz}{(1-x)^2} + \frac{zx}{(1-y)^2}}
$$
(proposed by N. Argilsan)
$$
\frac{\sqrt{xyz}}{x^2 + y^2 + z^2 - x^3 - y^3 - z^3} \le \sqrt{\frac{xy}{(1-z)^2} + \frac{yz}{(1-x)^2} + \frac{zx}{(1-y)^2}}
$$
(proposed by N. Argilsan)