IMC 2013 Problem 4

Let ${n \geqslant 3}$ and let ${{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define ${A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$ . Prove that: $${\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.$$

Proposed by Géza Kós, Eötvös University, Budapest.