1991 AMC 12 Problem 15

Attempt Problem 15 of the 1991 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 1991 AMC 12 solutions, or check the answer key.

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

15.

A circular table has exactly 6060 chairs around it. There are NN people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of NN is
1515
2020
3030
4040
5858
Answer: B
Concepts:circular arrangementcoveringextremal counting
Small hint:
No empty chair can have both neighboring chairs empty
Big hint:
Between consecutive occupied chairs there can be at most two empty chairs
Solution:
The condition says that every empty chair has an occupied neighbor, so no three consecutive chairs may all be empty. Therefore each occupied chair can account for at most itself and the two empty chairs following it, giving 60≤3N60\le3N and N≥20.N\ge20. Repeating the pattern occupied-empty-empty around the table attains N=20.N=20.
Thus the correct answer is B.
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