Difference between revisions of "1991 AHSME Problems/Problem 15"
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| + | == Problem == | ||
| + | |||
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>? | ||
| + | |||
| + | <math>\text{(A) } 15\quad | ||
| + | \text{(B) } 20\quad | ||
| + | \text{(C) } 30\quad | ||
| + | \text{(D) } 40\quad | ||
| + | \text{(E) } 58</math> | ||
| + | |||
| + | == Solution == | ||
| + | <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 == | ||
| + | {{AHSME box|year=1991|num-b=14|num-a=16}} | ||
| + | |||
| + | [[Category: Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 16:38, 23 February 2018
Problem
A circular table has 60 chairs around it. There are 
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 ?
Solution
If we fill every third chair with a person, then the condition is satisfied, giving 
. Decreasing any further means there is at least one gap of 
, so that the person can sit themselves in the middle (seat 
of ) and not be next to anyone. Hence the minimum value of is 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
