2010 AMC 12B Problems/Problem 8

Problem 8

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$ ^th and $64$ ^th, respectively. How many schools are in the city?

$\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26$

Solution

There are $x$ schools. This means that there are $3x$ people. Because no one's score was the same as another person's score, that means that there could only have been 1 median score. This implies that $x$ is an odd number. $x$ cannot be less than 23, because there wouldn't be a 64th place if there were. $x$ cannot be greater than 23 either, because that would tie Andrea and Beth. Thus, the only possible answer is $\boxed{23}$

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2010 AMC 12B Problems/Problem 8

Problem 8

Solution

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