Question 5
Prove the inequality
$$
\sum_{i\lt j} \frac{a_{i} a_{j}}{a_{i}+a_{j}} \leq \frac{n}{2\left(a_{1}+a_{2}+\cdots+a_{n}\right)} \sum_{i\lt j} a_{i} a_{j}
$$
for positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$.
$$
\sum_{i\lt j} \frac{a_{i} a_{j}}{a_{i}+a_{j}} \leq \frac{n}{2\left(a_{1}+a_{2}+\cdots+a_{n}\right)} \sum_{i\lt j} a_{i} a_{j}
$$
for positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$.