2013 IMO Shortlist N2

Prove that for any pair of positive integers $k$ and $n$ there exist $k$ positive integers $m_{1}, m_{2}, \ldots, m_{k}$ such that $$1+\frac{2^{k}-1}{n}=\left(1+\frac{1}{m_{1}}\right)\left(1+\frac{1}{m_{2}}\right) \cdots\left(1+\frac{1}{m_{k}}\right) .$$ (Japan)

证明对于任意一对正整数 $k$ 和 $n$，都存在 $k$ 个正整数 $m_{1}, m_{2}, \ldots, m_{k}$，使得 $$1+\frac{2^{k}-1}{n}=\left(1+\frac{1}{m_{1}}\right)\left(1+\frac{1}{m_{2}}\right) \cdots\left(1+\frac{1}{m_{k}}\right) 。$$(日本)