2010 AMC 10B Problem 17

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

Answer: B

Concepts:median (data)inequality logical deduction

Difficulty rating: 1600

Small hint:

If there are

n

schools, there are

3n

contestants

Big hint:

Andrea’s median place is

\dfrac{3n+1}{2}

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

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.