2010 AMC 12B Problem 8

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)parity inequality

Difficulty rating: 1490

Small hint:

With

n

schools there are

3n

students, and the median sits at position

\dfrac{3n+1}{2}

Big hint:

3n\ge64,

and Andrea placed ahead of Beth’s

37

th, so

\dfrac{3n+1}{2}\lt37

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.