2010 AMC 10B Problem 17

Let there be $n$ schools. Then there are $3n$ participants in the contest. Because the median is one student’s score, $3n$ is odd, so $n$ is odd. Carla’s $64$th place requires $3n\ge64,$ and hence the odd integer $n$ is at least $23.$ Andrea’s median place is $\dfrac{3n+1}{2}.$ Because she had the highest score on her team, she finished ahead of Beth, so $\dfrac{3n+1}{2}\lt37.$ This gives $3n\lt73,$ and therefore the odd integer $n$ is at most $23.$ Thus there are $23$ schools. Thus, B is the correct answer.