HMMT 二月 2003 · GEN1 赛 · 第 3 题

英文解析

3. There are 16 members on the Height-Measurement Matching Team. Each member wasasked, “How many other people on the team — not counting yourself — are exactlythe same height as you?” The answers included six 1’s, six 2’s, and three 3’s. Whatwas 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 sameheight, and then each of them would have given the answer 3. Thus, we need at leastfour 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.)