Question 42
Let $a$, $b$, $c \geq 0$ satisfy $a + b + c = 1$. Prove that
$$
\frac{\sqrt{a}}{b + 1} + \frac{\sqrt{b}}{c + 1} + \frac{\sqrt{c}}{a + 1} \gt \frac{1}{2}(\sqrt{a} + \sqrt{b} + \sqrt{c}) .
$$
$$
\frac{\sqrt{a}}{b + 1} + \frac{\sqrt{b}}{c + 1} + \frac{\sqrt{c}}{a + 1} \gt \frac{1}{2}(\sqrt{a} + \sqrt{b} + \sqrt{c}) .
$$