Let $n$ be an integer. Determine the largest constant $C$ possible such that for all $a_{1}, \ldots, a_{n} \geqslant 0$ we have $\left(\sum a_{i}\right)^{2} \geqslant C \sum_{i<j} a_{i} a_{j}$.
We add $2 \sum_{i<j} a_{i} a_{j}$ to the previous inequality, we find $C=\frac{2 n}{n-1}$.