Printable

2010 AMC 10B Problem 17

2010 AMC 10B Problem 17AMC 10 Step by Step

Name: Date: September 13, 2026

Statistics & Data · difficulty 3 of 5 · about 4 min · position P16-20

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?

A)
$22$
B)
$23$
C)
$24$
D)
$25$
E)
$26$

Answer

Key insight

With 3n students Andrea's median rank is (3n+1)/2, so n is odd; she is above 37th, so n < 25, and 64 students exist, so n >= 22.

Solution

Let there be $n$ schools, so $3n$ students with distinct scores. Andrea is the median, so she is a single middle student; that requires $3n$ to be odd, i.e. $n$ odd, and her rank is $\frac{3n+1}{2}$ .

Andrea outscored Beth, who is $37$ th, so
$\frac{3n+1}{2} < 37 \quad\Longrightarrow\quad 3n < 73 \quad\Longrightarrow\quad n \le 24.$

Carla is $64$ th, so there are at least $64$ students: $3n \ge 64$ , giving $n \ge 22$ .

The only odd integer in $22 \le n \le 24$ is $n = 23$ . Check: $69$ students, Andrea is $35$ th, above Beth's $37$ th, and Carla's $64$ th place exists.

The answer is $\boxed{\textbf{(B)}\ 23}$ .

Why this works

Every sentence becomes an inequality on $n$ : the median's rank gives an upper bound, the existence of a $64$ th place gives a lower bound, and parity (the median of distinct scores is an actual person only when the count is odd) selects the unique value. Squeeze from both sides, then apply the integrality condition.

The trap

Forgetting that a median of an even number of distinct scores is not a participant, which forces n to be odd.

Common mistakes

Forgetting that a median of an even number of distinct scores is not a participant, which forces n to be odd.
Using only the upper bound $n \le 24$ and picking $22$ or $24$ , or misreading "highest on her team" and allowing Andrea to be below $37$ th.

Techniques

Bound the quantity above/below or estimate to pin it down · Use an invariant, parity, or coloring argument