1990 AJHSME Problems/Problem 19

Problem

There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?

$\text{(A)}\ 30 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 41 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 119$

Solution

$p$ is a person seated, $o$ is an empty seat

The pattern of seating that results in the fewest occupied seats is $o<cmath>p</cmath>o<cmath>o</cmath>p<cmath>o</cmath>o<cmath>p</cmath>o$ $o$ ... $p$ $o$ we can group the seats in 3s $o<cmath>p</cmath>o$ $o$ p$$ (Error compiling LaTeX. Unknown error_msg)o$$ (Error compiling LaTeX. Unknown error_msg)o $p$ o $...$ o $p$ o $there are a total of$ \boxed{B}$ groups

1990 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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1990 AJHSME Problems/Problem 19

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