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 $37$ th and $64$ th, respectively. How many schools are in the city?

$22$

$23$

$24$

$25$

$26$

Solution:

With $n$ schools there are $3n$ students. Carla placed $64$ th, so $3n\ge64$ and $n\ge22.$

The scores are distinct and Andrea is the median, so $3n$ is odd, forcing $n$ odd and $n\ge23.$

Andrea's position is $\dfrac{3n+1}{2},$ and she beat Beth ( $37$ th), so $\dfrac{3n+1}{2}\lt37,$ giving $3n\lt73$ and $n\le24.$ The only odd value is $n=23.$

Thus, the correct answer is B.

2010 AMC 12B Problem 8

Solution:

Problem 8 in Other Years