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Difference between revisions of "1991 AHSME Problems/Problem 15"
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== Problem ==
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A circular table has 60 chairs around it. There are <math>N</math> 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 <math>N</math>?
 
A circular table has 60 chairs around it. There are <math>N</math> 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 <math>N</math>?
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<math>\text{(A) } 15\quad
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\text{(B) } 20\quad
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\text{(C) } 30\quad
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\text{(D) } 40\quad
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\text{(E) } 58</math>
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== Solution ==
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<math>\fbox{B}</math> If we fill every third chair with a person, then the condition is satisfied, giving <math>N=20</math>. Decreasing <math>N</math> any further means there is at least one gap of <math>4</math>, so that the person can sit themselves in the middle (seat <math>2</math> of <math>4</math>) and not be next to anyone. Hence the minimum value of <math>N</math> is <math>20</math>.
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== See also ==
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{{AHSME box|year=1991|num-b=14|num-a=16}} 
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[[Category: Introductory Combinatorics Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 16:38, 23 February 2018

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 $20$.

See also

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
All AHSME Problems and Solutions

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