Middle European Mathematical Olympiad 2011 Problem T-2

Let $a, b, c$ be positive real numbers such that $$\frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c} = 2.$$ Prove that $$\frac{\sqrt{a} + \sqrt{b} + \sqrt{c}}{2} \geqslant \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}.$$