RO-TST 2012 Problem 10

Let $m$ and $n$ be two positive integers greater than $1$ . Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that $$\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}$$