ISL 2006 A4

Prove the inequality $$\sum_{i<j} \frac{a_{i} a_{j}}{a_{i}+a_{j}} \leq \frac{n}{2\left(a_{1}+a_{2}+\cdots+a_{n}\right)} \sum_{i<j} a_{i} a_{j}$$ for positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$. (Serbia)