HMMT 二月 2022 · 冲刺赛 · 第 33 题

HMMT February 2022 — Guts Round — Problem 33

[20] In last year’s HMMT Spring competition, 557 students submitted at least one answer to each of the three individual tests. Let S be the set of these students, and let P be the set containing the 30 problems on the individual tests. Estimate A , the number of subsets R ⊆ P for which some student in S answered 2 the questions in R correctly but no others. An estimate of E earns max(0 , ⌊ 20 − | A − E |⌋ ) points. 3

解析

[20] In last year’s HMMT Spring competition, 557 students submitted at least one answer to each of the three individual tests. Let S be the set of these students, and let P be the set containing the 30 problems on the individual tests. Estimate A , the number of subsets R ⊆ P for which some student in 2 S answered the questions in R correctly but no others. An estimate of E earns max(0 , ⌊ 20 − | A − E |⌋ ) 3 points. Proposed by: Daniel Zhu Answer: 450