HMMT 十一月 2014 · THM 赛 · 第 3 题
HMMT November 2014 — THM Round — Problem 3
题目详情
- 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
- 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 is n ! p = . n (2 n + 1)!! Clearly this is decreasing in n . It is easy to see that 1 · 2 · 3 · 4 · 5 8 p = = > 0 . 01 5 3 · 5 · 7 · 9 · 11 693 and 6 48 p = p = < 0 . 01 . 6 5 13 693 · 13 1 Hence the answer is n = 6. Heuristically, p ≈ p for each n , so arriving at these estimates for the n +1 n 2 correct answer of n is not difficult.