1990 AJHSME Problems/Problem 19

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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

If there are $120/2=60$ seats occupied, then it is possible to get an occupation of the seats where no person sits next to someone else. However, the only way is through alternating seats.

Thus, if another person joins, two people must sit next to each other, so $\boxed{\text{D}}$ .

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

From AoPSWiki

Problem

Solution

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