HMMT 十一月 2014 · THM 赛 · 第 3 题
HMMT November 2014 — THM Round — Problem 3
题目详情
英文原题
- Find the smallest positive integer n such that, if there are initially n + 1 townspeople and n goons,
then the probability the townspeople win is less than 1%.
解析
英文解析
- Find the smallest positive integer n such that, if there are initially n + 1 townspeople and n goons,
then the probability the townspeople win is less than 1%.
Answer: 6 By a similar inductive argument, the probability for a given n isn !
p = .
(2 n + 1)!!n
Clearly this is decreasing in n . It is easy to see that
1 · 2 · 3 · 4 · 5 8
p = = > 0 . 01
3 · 5 · 7 · 9 · 11 6935
6 48 andp = p = < 0 . 01 .
6 5
13 693 · 13
Hence the answer is n = 6. Heuristically, p ≈ p for each n , so arriving at these estimates for the 1
n +1 ncorrect answer of n is not difficult.2