RO-NMO 2006 Problem 102

Prove that for all $a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $n \in \mathbb N^\ast$ we have $$\frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} .$$