Track / Stage 4 / 11 of 340 #271 of 2000

Problem 271

AMC 12 late, AIME early

Combinatorics Difficulty 4.4 Find the answer Harvard-MIT Mathematics Tournament · United States

Problem:

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.)

Your answer

Official solution

Solution:

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 $n+1$ .)

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