APMO 2012 Problem 5

Let $n$ be an integer greater than or equal to $2$ . Prove that if the real numbers $a_1 , a_2 , \cdots , a_n$ satisfy $a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n$ , then $$\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2}$$ must hold.