Competition problem — Combinatorics

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Combinatorics Difficulty 4.4 AIME Find the answer 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

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.