IMC 2013 · Problem 4

Let $n \ge 3$ and let $x_1, \dots, x_n$ be nonnegative real numbers. Define $A = \sum\limits_{i=1}^{n} x_i$, $B = \sum\limits_{i=1}^{n} x_i^2$ and $C = \sum\limits_{i=1}^{n} x_i^3$. Prove that

$$(n+1)A^2B + (n-2)B^2 \ge A^4 + (2n-2)AC.$$