Let $n$ be an integer. Determine the largest constant $C$ possible so that for all $a_{1} \ldots, a_{n} \geqslant 0$ we have $\sum a_{i}^{2} \geqslant C \sum_{i<j} a_{i} a_{j}$.
The inequality between the arithmetic mean and the quadratic mean is written as $\left(a_{1}+\cdots+a_{n}\right)^{2} \leqslant n\left(a_{1}^{2}+\cdots+a_{n}^{2}\right)$, or equivalently $\sum a_{i}^{2}+2 \sum_{i<j} a_{i} a_{j} \leqslant n \sum a_{i}^{2}$. Therefore, we have $C=\frac{2}{n-1}$.