Question 90
Prove that for all positive numbers $x, y, z$ such that $x + y + z = 1$ the following inequality holds
$$
\frac{x^{2}}{1 + y} + \frac{y^{2}}{1 + z} + \frac{z^{2}}{1 + x} \leq 1
$$
$$
\frac{x^{2}}{1 + y} + \frac{y^{2}}{1 + z} + \frac{z^{2}}{1 + x} \leq 1
$$