APMO 2012 - Asian Pacific Mathematics Olympiad Question 5

Let $n\ge2$ be an integer. Prove that if real numbers $a_1,\ldots,a_n$ satisfy $a_1^2+\cdots+a_n^2=n$, then
$$\sum_{1\le i\lt j\le n}\frac1{n-a_i a_j}\le\frac n2.$$