IMO Shortlist 2006 A4

[Phlembac Adib Hasan]

Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} +\cdots + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.

Proposed by Dusan Dukic, Serbia