. Let $n$ be a natural number, find the smallest constant $C$ such that for all $x_{1}, \ldots, x_{n} \geqslant 0$

$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant C\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
. We start from the following inequality:

$$
\left(\sum_{i=1}^{n} x_{i}\right)^{4}=\left(\sum_{1 \leqslant i<j \leqslant n} x_{i}^{2}+2 x_{i} x_{j}\right)^{2} \geqslant 8\left(\sum_{1 \leqslant k \leqslant n} x_{k}^{2}\right)\left(\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\right)
$$

which follows from the arithmetic-geometric mean inequality. Given that for any pair of indices $i \neq j$, $\sum_{1 \leqslant k \leqslant n} x_{k}^{2} \geqslant x_{i}^{2}+x_{j}^{2}$ with equality if and only if all $x_{k}$ with $k \neq i$ and $k \neq j$ are zero, we obtain that

$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant \frac{1}{8}\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$

with equality if and only if at most 2 terms are non-zero and they are equal. The constant sought is therefore $C=1 / 8$.