Question 47
(ROM 6) Prove the inequality
$$x_1x_2\cdots x_k\left(x_1^{n-1}+x_2^{n-1}+\cdots+x_k^{n-1}\right)\le x_1^{n+k-1}+x_2^{n+k-1}+\cdots+x_k^{n+k-1},$$
where $x_i>0$ $(i=1,2,\ldots,k)$, $k\in\mathbb N$, $n\in\mathbb N$.
$$x_1x_2\cdots x_k\left(x_1^{n-1}+x_2^{n-1}+\cdots+x_k^{n-1}\right)\le x_1^{n+k-1}+x_2^{n+k-1}+\cdots+x_k^{n+k-1},$$
where $x_i>0$ $(i=1,2,\ldots,k)$, $k\in\mathbb N$, $n\in\mathbb N$.