Question

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?

$30$

$40$

$41$

$60$

$119$

Solution:

To force the next person next to someone, every empty seat must be adjacent to an occupied one. The most efficient way is to seat people in a repeating pattern of empty-occupied-empty, filling the middle seat of every group of three.

With $120$ seats, this uses $\dfrac{120}{3} = 40$ occupied seats.

Thus, the correct answer is B .

1990 AMC 8 Problem 19

Solution:

Problem 19 in Other Years