AMC 10 · 2010 · #17

Grade 6 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

mean-median-mode-rangelogical-deductionlinear-equations-one-var bound-inequality-then-enumerate ↑ Prerequisites: mean-median-mode-range

📏 Long solution 💡 3 insights

Problem

Every high school in the city of Euclid sent a team of 3 students to a math contest, so the total number of students is 3 times the number of schools. Each participant received a different score, so every rank is unique. Andrea's score was the median among all students, and hers was the highest score on her team. Her teammates Beth and Carla placed 37th and 64th. How many schools are in the city?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Name the number of schools with a letter so the total count and Andrea's median rank become simple expressions. The two teammate placements act as boundaries: they squeeze the median rank from above and below (Extreme Principle). Because the total must be odd and the schools count is a whole number, only one choice survives (Eliminate Possibilities).

1STEP 1

Name the number of schools

Let n be the number of schools. Each school sends 3 students, so there are 3n students in all.

total students = 3n

2STEP 2

Locate the median rank

A single median needs an odd count, so 3n and n are both odd, and Andrea sits at rank (3n+1)/2.

Andrea's rank = (3n+1)/2, 3n odd → n odd

3STEP 3

Use Beth to bound n above

Andrea beats Beth's 37th, so her rank is at most 36; then (3n+1)/2 ≤ 36 gives n ≤ 23.

(3n+1)/2 ≤ 36 → 3n ≤ 71 → n ≤ 23

4STEP 4

Use Carla to bound n below

Carla's 64th place exists, so 3n ≥ 64; since 3 times 21 is only 63, n ≥ 22.

3n ≥ 64 → n ≥ 22

5STEP 5

Keep only the value that fits

So 22 ≤ n ≤ 23 with n odd, and only 23 is odd in that range, so n = 23.

22 ≤ n ≤ 23, n odd → n = 23

6STEP 6

Confirm the count

23 schools give 69 students, an odd count, so Andrea is 35th — ahead of Beth at 37th and Carla at 64th.

3 · 23 = 69, (69+1)/2 = 35

Answer

Check the answer against every clue. 23 schools give 69 students, an odd count, so a single median exists at rank 35. Andrea at 35th beats both teammates (37th and 64th), so she is her team's top scorer, and 64th place exists because 69 is at least 64. All conditions are met, and 23 is one of the listed choices.

💡Key takeaway

Name the unknown, turn each clue into a squeeze from above and below, then keep only the whole number with the right parity.

Name the number of schools
Locate the median rank
Use Beth to bound n above
Use Carla to bound n below
Keep only the value that fits
Confirm the count