HMMT 二月 2006 · 冲刺赛 · 第 43 题

HMMT February 2006 — Guts Round — Problem 43

Write down at least one, and up to ten, different 3-digit prime numbers. If you somehow fail to do this, we will ignore your submission for this problem. Otherwise, you’re entered into a game with other teams. In this game, you start with 10 points, and each number you write down is like a bet: if no one else writes that number, you gain 1 point, but if anyone else writes that number, you lose 1 point. Thus, your score on this problem can be anything from 0 to 20.

解析

Write down at least one, and up to ten, different 3-digit prime numbers. If you somehow fail to do this, we will ignore your submission for this problem. Otherwise, you’re entered into a game with other teams. In this game, you start with 10 points, and each number you write down is like a bet: if no one else writes that number, you gain 1 point, but if anyone else writes that number, you lose 1 point. Thus, your score on this problem can be anything from 0 to 20. Solution: There are 143 three-digit primes. None of the following necessarily applies to the actual contest, but it might be useful to think about. Suppose that you’re trying to maximize your expected score on this problem. Then you should write down a number if you think the probability that someone else is writing it is less than 1 / 2 (of course, limit yourself to 10 numbers). You should avoid writting down any number if you think the probability that someone else is writing it is more than 1 / 2 (of course, write down at least 1 number). Suppose that you expect a total of M different numbers are going to be written down, but have no idea what numbers they might be. If you think M ≥ 72, you should write down 10 numbers at random; if M ≤ 71, you should write just 1 number.