2016 IMO Shortlist N4

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^{k}+m n^{l}+1$ divides $n^{k+l}-1$. Prove that - $m=1$ and $l=2 k$; or - $l \mid k$ and $m=\frac{n^{k-l}-1}{n^{l}-1}$.

令 $n、m、k$ 和 $l$ 为正整数，且 $n \neq 1$ 使得 $n^{k}+m n^{l}+1$ 除以 $n^{k+l}-1$。证明 - $m=1$ 且 $l=2 k$；或 - $l \mid k$ 和 $m=\frac{n^{k-l}-1}{n^{l}-1}$。