RO-NMO 2009 Problem 93

Let be a natural number $n,$ a permutation $\sigma$ of order $n,$ and $n$ nonnegative real numbers $a_1,a_2,\ldots , a_n.$ Prove the following inequality. $$\left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right)$$