2023/24 AHSMC Part 1 Problem 1

Problem 1 of 16

In a competition, the participants are assigned a positive integer score on a scale from $1$ to $10$ (inclusive). The minimum number of participants required to guarantee that at least three of them achieve the same score is

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Solution

If the number of participants is at most $20$ then each score from $1$ to $10$ could be assigned to at most two participants. If the number of participants is $21$ , then by the pigeonhole principle some score between $1$ and $10$ is assigned to at least three participants. Therefore the minimum number required is $21$ . The answer is (C).

University of Alberta

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