Question 15
Let $a_i\gt0$ for $i=1,\ldots,n$ and $\sum_{i=1}^n a_i=1$. Prove that for every positive integer $k$,
$$\prod_{i=1}^n\left(a_i^k+\frac1{a_i^k}\right)\ge\left(n^k+\frac1{n^k}\right)^n.$$
$$\prod_{i=1}^n\left(a_i^k+\frac1{a_i^k}\right)\ge\left(n^k+\frac1{n^k}\right)^n.$$