Let
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
be a fixed integer. Find the least constant
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
such the inequality
holds for any
![x_{1}, \ldots ,x_{n} \geq 0](/media/m/c/b/e/cbe247699e65710f73c894bed931f32e.png)
(the sum on the left consists of
![\binom{n}{2}](/media/m/c/c/c/ccc4290a75cc28069badae7324b13fe8.png)
summands). For this constant
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, characterize the instances of equality.
%V0
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
$$\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4$$
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.