ILL 1989 Problem 61

Prove for $0 < k \leq 1$ and $a_i \in \mathbb{R}^+,$ $i = 1,2 \ldots, n$ the following inequality holds:

$$\left( \frac{a_1}{a_2 + \ldots + a_n} \right)^k + \ldots + \left( \frac{a_n}{a_1 + \ldots + a_{n-1}} \right)^k \geq \frac{n}{(n-1)^k}.$$