HMMT 二月 2018 · 冲刺赛 · 第 9 题

HMMT February 2018 — Guts Round — Problem 9

[ 7 ] In a game, N people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let m be the maximum possible number such that it is possible to win the game by writing down m . Find the smallest possible value of N for which it is possible to win the game by writing down m in a room of N people.

解析

[ 7 ] In a game, N people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let m be the maximum possible number such that it is possible to win the game by writing down m . Find the smallest possible value of N for which it is possible to win the game by writing down m in a room of N people. Proposed by: Kevin Sun Answer: 34 Since the average of the numbers is at most 100, the winning number is an integer which is at most two-thirds of 100, or at most 66. This is achieved in a room with 34 people, in which 33 people pick 100 and one person picks 66, so the average number is 99. Furthermore, this cannot happen with less than 34 people. If the winning number is 66 and there are N people, the sum of the numbers must be 99. then we must have that 99 N ≤ 66 + 100( N − 1), which reduces to N ≥ 34.