Question

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 $37^\text{th}$ and $64^\text{th},$ respectively. How many schools are in the city?

$22$

$23$

$24$

$25$

$26$

Solution:

Let there be $x$ schools. Then there are $3x$ participants in the contest.

Since everyone has a unique score, there is also a unique median. This means $x$ is odd.

Note there are at least $23$ teams, since otherwise a $64^\text{th}$ placed wouldn't exist.

We also have at most $23$ teams, since otherwise we have Andrea's place as being greater than Beth's.

This means that there are $23$ teams.

Thus, B is the correct answer.

2010 AMC 10B Problem 17

Solution:

Problem 17 in Other Years