Question 2
Find the least constant $C$ such that for any integer $n \gt 1$ the inequality $$
\sum_{1 \le i \lt j \le n} x_i x_j (x_i^2 + x_j^2)
\le C \left( \sum_{1 \le i \le n} x_i \right)^4
$$ holds for all real numbers $x_1, \dots, x_n \ge 0$. Determine the cases of equality.
\sum_{1 \le i \lt j \le n} x_i x_j (x_i^2 + x_j^2)
\le C \left( \sum_{1 \le i \le n} x_i \right)^4
$$ holds for all real numbers $x_1, \dots, x_n \ge 0$. Determine the cases of equality.