1991 AHSME Problems/Problem 15

Problem

A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$ ?

$\text{(A) } 15\quad \text{(B) } 20\quad \text{(C) } 30\quad \text{(D) } 40\quad \text{(E) } 58$

Solution

$\fbox{B}$ If we fill every third chair with a person, then the condition is satisfied, giving $N=20$ . Decreasing $N$ any further means there is at least one gap of $4$ , so that the person can sit themselves in the middle (seat $2$ of $4$ ) and not be next to anyone. Hence the minimum value of $N$ is $20$ .

1991 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1991 AHSME Problems/Problem 15

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