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, 2013

Algebra Difficulty 5.4 AIME, harder Prove it United States

Problem:

Marty and three other people took a math test. Everyone got a non-negative integer score. The average score was $20$ . Marty was told the average score and concluded that everyone else scored below average. What was the minimum possible score Marty could have gotten in order to definitively reach this conclusion?

Solution

Solution:

$61$

Suppose for the sake of contradiction Marty obtained a score of $60$ or lower. Since the mean is $20$ , the total score of the $4$ test takers must be $80$ . Then there exists the possibility of $2$ students getting $0$ , and the last student getting a score of $20$ or higher. If so, Marty could not have concluded with certainty that everyone else scored below average.

With a score of $61$ , any of the other three students must have scored points lower or equal to $19$ points. Thus Marty is able to conclude that everyone else scored below average.

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