1976 AMC 12 Problem 12

Attempt Problem 12 of the 1976 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1976 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples?

$4$

$5$

$6$

$24$

$25$

Answer: C

Concepts:pigeonhole principle basic counting

Difficulty rating: 1380

Small Hint:

Count the possible apple totals from $120$ through $144$ , inclusive

Big Hint:

If every total occurred at most five times, there would be at most $25\cdot5$ crates

Solution:

There are $144-120+1=25$ possible apple counts. Since $128>25\cdot5,$ some count occurs at least $6$ times. This is sharp: distribute $128$ crates so that three counts occur $6$ times and the other twenty-two occur $5$ times. Thus the largest guaranteed $n$ is $6.$

Therefore, the correct answer is C.

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Full Exam

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