Question 65
Determine all natural numbers $n$ such that for arbitrary real numbers $x_{1}, \ldots, x_{n}$ the following holds:
$$
\left(\frac{x_{1}^{n}+\ldots+x_{n}^{n}}{n}-x_{1} \cdot \ldots \cdot x_{n}\right)\left(x_{1}+\ldots+x_{n}\right) \geq 0
$$
$$
\left(\frac{x_{1}^{n}+\ldots+x_{n}^{n}}{n}-x_{1} \cdot \ldots \cdot x_{n}\right)\left(x_{1}+\ldots+x_{n}\right) \geq 0
$$