Question 145
Let $a$, $b$ and $c$ be positive real numbers. Prove that
$$
\frac{\sqrt{a+b+c}+\sqrt{a}}{b+c} + \frac{\sqrt{a+b+c}+\sqrt{b}}{c+a} + \frac{\sqrt{a+b+c}+\sqrt{c}}{a+b} \ge \frac{9+3\sqrt{3}}{2\sqrt{a+b+c}}
$$
$$
\frac{\sqrt{a+b+c}+\sqrt{a}}{b+c} + \frac{\sqrt{a+b+c}+\sqrt{b}}{c+a} + \frac{\sqrt{a+b+c}+\sqrt{c}}{a+b} \ge \frac{9+3\sqrt{3}}{2\sqrt{a+b+c}}
$$