Question

设集合 $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}$ 。 Susan 按如下方式列出一个表：对于 $\mathcal{S}$ 的每个二元子集，她在表上写下该子集中较大的元素。求表上所有数的和。

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

答案

答案：484

知识点：数对计数双重计数

难度评级：1840

解答：

元素 $x$ 与集合中每个比它小的元素组成二元子集时，都会成为较大的元素，因此 $x$ 对总和的贡献次数等于比它小的元素个数。将集合排序为 $1, 2, 3, 5, 8, 13, 21, 34$ ，表上所有数的和为 $\begin{aligned} &0(1) + 1(2) + 2(3) + 3(5) \\ &\quad {}+ 4(8) + 5(13) + 6(21) \\ &\quad {}+ 7(34) \\ &= 2 + 6 + 15 + 32 \\ &\quad {}+ 65 + 126 + 238 = 484. \end{aligned}$

An element $x$ is the greater element of a two-element subset exactly once for each smaller element of the set, so $x$ contributes to the sum once per element below it. Sorting the set as $1, 2, 3, 5, 8, 13, 21, 34,$ the sum of the list is $\begin{aligned} &0(1) + 1(2) + 2(3) + 3(5) \\ &\quad {}+ 4(8) + 5(13) + 6(21) \\ &\quad {}+ 7(34) \\ &= 2 + 6 + 15 + 32 \\ &\quad {}+ 65 + 126 + 238 = 484. \end{aligned}$

2003 AIME I 第 3 题

答案：484

难度评级：1840

解答：

其他年份的第 3 题