2013 IMO 第 1 题

Let $k$ and $n$ be positive integers. Prove that there exist positive integers $m_{1}$ , ..., $m_{k}$ such that

$$1 + \frac{2^{k} - 1}{n} = \left(1 + \frac{1}{m_{1}}\right)\left(1 + \frac{1}{m_{2}}\right)\dots \left(1 + \frac{1}{m_{k}}\right).$$

令 $k$ 和 $n$ 为正整数。证明存在正整数 $m_{1}$ , ..., $m_{k}$ 使得

$$1 + \frac{2^{k} - 1}{n} = \left(1 + \frac{1}{m_{1}}\right)\left(1 + \frac{1}{m_{2}}\right)\dots \left(1 + \frac{1}{m_{k}}\right)。$$