HMMT 二月 2002 · 冲刺赛 · 第 26 题

HMMT February 2002 — Guts Round — Problem 26

[ ± 4] Another professor enters the same room and says, “Each of you has to write down an integer between 0 and 200. I will then compute X , the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number closest to X (either above or below X ) will receive a prize.” One student, who misunderstood the question, announces to the class that he will write the number 107. If among the other 99 students it is common knowledge that all 99 of them will write down the best response, and there is no further communication between students, what single integer should each of the 99 students write down?

解析

Another professor enters the same room and says, “Each of you has to write down an integer between 0 and 200. I will then compute X , the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number closest to X (either above or below X ) will receive a prize.” One student, who misunderstood the question, announces to the class that he will write the number 107. If among the other 99 students it is common knowledge that all 99 of them will write down the best response, and there is no further communication between students, what single integer should each of the 99 students write down? Solution: Use the same logic to get 7 . Note 6 and 8 do not work.