AMC 10 · 2011 · #11

Grade 6 arithmetic
Archetype Modular Cycle Frequency Count └ Sub-archetype Floor Count Across Residue Bins
pigeonhole-principlefloor-function extremal-construction ↑ Prerequisites: floor-function
📏 Medium solution 💡 2 insights
Problem
In a room of 52 people, every birthday lands in one of the 12 months of the year. The sentence "at least n people in this room share the same birth month" must stay true no matter how the birthdays happen to fall. Find the largest such n.

Pick an answer.

(A)
2
(B)
3
(C)
4
(D)
5
(E)
12

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

"Guaranteed no matter what" is a signal for Tool #14 (Extreme Principle): imagine the arrangement that tries hardest to keep every month small, and see what the fullest month is forced to reach even then. The fairest split is found by dividing the people as evenly as possible across the months. Tool #3 (Eliminate Possibilities) closes the case by showing that a valid arrangement exists whose fullest month is exactly this number, so nothing larger can be guaranteed.

1STEP 1

Aim for the fairest split

Picture the arrangement that fights the statement hardest: the 52 birthdays spread as evenly as possible over the 12 months.

52 people ⟶ 12 months, as evenly as possible
2STEP 2

Divide people among months

Divide 52 by 12: every month takes 4 people, using 12 × 4 = 48, and 4 are left over.

52 ÷ 12 = 4 remainder 4 (12 × 4 = 48, 52 - 48 = 4)
3STEP 3

Place the leftovers

Each leftover raises a month from 4 to 5, so even the fairest split forces some month to hold at least 5 people.

4 months → 5 people, 8 months → 4 people
4STEP 4

Check that six is not forced

But 8 × 4 + 4 × 5 = 52 is a legal split whose fullest month holds only 5, so 6 is never forced and the answer is 5, choice (D).

8 × 4 + 4 × 5 = 32 + 20 = 52 → max = 5 → (D)
Answer
5
The average number of people per month is 52 ÷ 12 ≈ 4.3, so the busiest month must be at least the next whole number up, which is 5. That matches the answer. The choices bracket it sensibly: 2, 3, 4 are too weak to be the largest guarantee, and 12 would require far more than 52 people to force 12 birthdays into a single month, which is impossible.
💡Key takeaway

Spread things as evenly as you can; the leftovers force the fullest bin up, so 52 people over 12 months always crowd 5 into one month.

  • Aim for the fairest split
  • Divide people among months
  • Place the leftovers
  • Check that six is not forced

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