2012 IMO 第 6 题

Find all positive integers $n$ for which there exist non-negative integers $a_{1}, a_{2}, \ldots , a_{n}$ such that

$$\frac{1}{2^{a_{1}}} + \frac{1}{2^{a_{2}}} + \dots + \frac{1}{2^{a_{n}}} = \frac{1}{3^{a_{1}}} + \frac{2}{3^{a_{2}}} + \dots + \frac{n}{3^{a_{n}}} = 1.$$

查找存在非负整数 $a_{1}, a_{2}, \ldots , a_{n}$ 的所有正整数 $n$ ，使得

$$\frac{1}{2^{a_{1}}} + \frac{1}{2^{a_{2}}} + \dots + \frac{1}{2^{a_{n}}} = \frac{1}{3^{a_{1}}} + \frac{2}{3^{a_{2}}} + \dots + \frac{n}{3^{a_{n}}} = 1.$$