BG-MO 2009 Problem 6

Prove that if $a_{1},a_{2},\ldots,a_{n}$ , $b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $c_{1},c_{2},\ldots,c_{n}$ are positive real numbers, than $\left(\sum_{i,j = 1}^{n}\frac {a_{i}a_{j}}{c_{i} + c_{j}}\right)\left(\sum_{i,j = 1}^{n}\frac {b_{i}b_{j}}{c_{i} + c_{j}}\right)\ge \left(\sum_{i,j = 1}^{n}\frac {a_{i}b_{j}}{c_{i} + c_{j}}\right)^{2}$ .