HMMT 二月 2021 · 冲刺赛 · 第 11 题

HMMT February 2021 — Guts Round — Problem 11

[10] For each positive integer 1 ≤ m ≤ 10, Krit chooses an integer 0 ≤ a < m uniformly at random. m Let p be the probability that there exists an integer n for which n ≡ a (mod m ) for all m . If p can be m a written as for relatively prime positive integers a and b , compute 100 a + b . b 3

解析

[10] For each positive integer 1 ≤ m ≤ 10, Krit chooses an integer 0 ≤ a < m uniformly at random. m Let p be the probability that there exists an integer n for which n ≡ a (mod m ) for all m . If p can m a be written as for relatively prime positive integers a and b , compute 100 a + b . b Proposed by: Daniel Zhu Answer: 1540 Solution: Tuples of valid a correspond with residues mod lcm(1 , 2 , . . . , 10), so the answer is m 3 2 lcm(1 , 2 , . . . , 10) 2 · 3 · 5 · 7 1 = = . 8 4 2 10! 2 · 3 · 5 · 7 1440 3