2010 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra Number Theory

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?

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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.

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