HMMT 二月 2005 · COMB 赛 · 第 2 题
HMMT February 2005 — COMB Round — Problem 2
题目详情
- How many nonempty subsets of { 1 , 2 , 3 , . . . , 12 } have the property that the sum of the largest element and the smallest element is 13?
解析
- How many nonempty subsets of { 1 , 2 , 3 , . . . , 12 } have the property that the sum of the largest element and the smallest element is 13? Solution: 1365 If a is the smallest element of such a set, then 13 − a is the largest element, and for the remaining elements we may choose any (or none) of the 12 − 2 a elements 12 − 2 a a + 1 , a + 2 , . . . , (13 − a ) − 1. Thus there are 2 such sets whose smallest element is a . Also, 13 − a ≥ a clearly implies a < 7. Summing over all a = 1 , 2 , . . . , 6, we get a total of 10 8 6 0 5 4 0 6 2 + 2 + 2 + · · · + 2 = 4 + 4 + · · · + 4 = (4 − 1) / (4 − 1) = 4095 / 3 = 1365 possible sets.