Question 31
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove that the following inequality holds
$$
\frac{1}{2}(\sqrt{a} + \sqrt{b} + \sqrt{c}) + \frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \ge 3
$$
When does equality hold?
$$
\frac{1}{2}(\sqrt{a} + \sqrt{b} + \sqrt{c}) + \frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \ge 3
$$
When does equality hold?