HMMT 十一月 2020 · 冲刺赛 · 第 8 题

HMMT November 2020 — Guts Round — Problem 8

[7] A small village has n people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If 10 ≤ n ≤ 100, what is the number of possible values of n ?

解析

[7] A small village has n people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If 10 ≤ n ≤ 100, what is the number of possible values of n ? Proposed by: Vincent Bian Answer: 61 ( ) n Solution: The problem asks for the number of n that divide , which happens exactly when 3 ( n − 1)( n − 2) is an integer. Regardless of the parity of n , ( n − 1)( n − 2) is always divisible by 2. Also, 2 · 3 ( n − 1)( n − 2) is divisible by 3 if and only if n is not a multiple of 3. Of the 91 values from 10 to 100, 30 are divisible by 3, so our answer is 61.