HMMT 二月 2006 · 冲刺赛 · 第 12 题
HMMT February 2006 — Guts Round — Problem 12
题目详情
英文原题
- [7] For each positive integer n let S denote the set { 1 , 2 , 3 , . . . , n } . Compute the numbernof triples of subsets A, B, C of S (not necessarily nonempty or proper) such that A is a
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subset of B and S − A is a subset of C .
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IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUNDth
The problems in this batch all depend on each other. If you solve them correctly, you willproduce a triple of mutually consistent answers. There is only one such triple. Your scorewill be determined by how many of your answers match that triple.
解析
英文解析
- For each positive integer n let S denote the set { 1 , 2 , 3 , . . . , n } . Compute the numbernof triples of subsets A, B, C of S (not necessarily nonempty or proper) such that A
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is a subset of B and S − A is a subset of C .
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4012
Answer: 2
Solution: Let A , B , C be sets satisfying the said conditions. Note that 1 ∈ Ao o o oimplies that 1 ∈ B and 1 ∈ / S − A so that 1 may or may not be in C . Also,
o 2006 o o
1 ∈ / A implies that 1 ∈ S − A ⊂ C while 1 may or may not be in B . Thus 3
o 2006 o o othere are four possibilities for the distribution of 1, and since the same argument holds
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independently for 2, 3, . . . , 2006, the answer is 4 or 2 .