ILL 1983 Problem 49

Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$ , prove that $$n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).$$