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?

$22$

$23$

$24$

$25$

$26$

Answer: B

Concepts:median (data)inequality logical deduction

Difficulty rating: 1600

Solution:

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 $64^\text{th}$ 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.

2010 AMC 10B Problem 17

Answer: B

Difficulty rating: 1600

Solution:

Problem 17 in Other Years