1991 USAMO 第 2 题

For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that

$$\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),$$

where "$\Sigma$" denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.

对于任何非空数字集合 $\,S\,$，设 $\,\sigma(S)\,$ 和 $\,\pi(S)\,$ 分别表示 $\,S\,$ 元素的和与积。证明

$$\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),$$

其中“$\Sigma$”表示涉及$\{1,2,3,\ldots,n\}$的所有非空子集$S$的总和。