HMMT 二月 2006 · 冲刺赛 · 第 43 题

英文解析

43. Write down at least one, and up to ten, different 3-digit prime numbers. If you somehowfail to do this, we will ignore your submission for this problem. Otherwise, you’reentered into a game with other teams. In this game, you start with 10 points, andeach 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 onthis problem can be anything from 0 to 20.
    Solution: There are 143 three-digit primes. None of the following necessarily appliesto the actual contest, but it might be useful to think about. Suppose that you’retrying to maximize your expected score on this problem. Then you should write downa 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 numberif 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 numbersare going to be written down, but have no idea what numbers they might be. If youthink M ≥ 72, you should write down 10 numbers at random; if M ≤ 71, you shouldwrite just 1 number.