2010 AMC 10B Problems/Problem 17

Problem

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

Let the $n$ be the number of schools, $3n$ be the number of contestants, and $x$ be Andrea's score. Since the number of participants divided by three is the number of schools, $n\geq\frac{64}3$ . $n$ is an integer, so $n\geq22$ . Since Andrea received a higher score than her teammates, $x\leq36$ . Since $36$ is the maximum possible median, then $2*36-1=71$ is the maximum possible number of participants. Therefore, $3n\leq71\Rightarrow n\leq\frac{71}3=23\frac23$ . This yields the compound inequality: $22\leq n\leq 23\frac23$ . Since a set with an even number of elements has a median that is the average of the two middle terms, an occurrence that cannot happen in this situation, $n$ cannot be even. $\boxed{\mathrm {(B)} 23}$ is the only other option.

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2010 AMC 10B Problems/Problem 17

Problem

Solution