Question 15
Let $n$ be a positive integer and let $x_1, x_2, \dots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that
$$
\sum_{i=1}^{n} x_{i}^{n}(1 + x_{i}) \geq \frac{n}{2^{n-1}} \prod_{i=1}^{n}(1 + x_{i}).
$$
$$
\sum_{i=1}^{n} x_{i}^{n}(1 + x_{i}) \geq \frac{n}{2^{n-1}} \prod_{i=1}^{n}(1 + x_{i}).
$$