Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied $$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$
Source: Swiss IMO TST 2016. Problem 4
Tags: inequalities
Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied $$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$