HMMT 二月 2003 · GEN1 赛 · 第 3 题

HMMT February 2003 — GEN1 Round — Problem 3

There are 16 members on the Height-Measurement Matching Team. Each member was asked, “How many other people on the team — not counting yourself — are exactly the same height as you?” The answers included six 1’s, six 2’s, and three 3’s. What was the sixteenth answer? (Assume that everyone answered truthfully.)

解析

There are 16 members on the Height-Measurement Matching Team. Each member was asked, “How many other people on the team — not counting yourself — are exactly the same height as you?” The answers included six 1’s, six 2’s, and three 3’s. What was the sixteenth answer? (Assume that everyone answered truthfully.) Solution: 3 For anyone to have answered 3, there must have been exactly 4 people with the same height, and then each of them would have given the answer 3. Thus, we need at least four 3’s, so 3 is the remaining answer. (More generally, a similar argument shows that the number of members answering n must be divisible by n + 1.)