AMC 8 1990 Problem 19
AMC8 1990 年第 19 题
鸽巢原理与策略★★★☆☆
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?
一排有 120 个座位。最少需要坐满多少个座位，才能使下一个入座的人必须挨着某人坐？
- A.30
- B.40
- C.41
- D.60
- E.119
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
A qualifying arrangement with 40 occupied seats can be created by dividing the 120 -seat row into 40 groups of 3 adjacent seats and placing a person in the middle seat of each group.
It is impossible to find a qualifying arrangement with 39 or fewer occupied seats. Suppose the row of seats is initially empty, and each person who sits down must choose an empty seat that is not next to an occupied seat. Then each person who sits down will eliminate at most 3 available seats from people who sit down later. Therefore, after 39 people sit down, there must be at least available seats, and thus a qualifying arrangement with 39 or fewer occupied seats does not exist.
Therefore, a qualifying arrangement has at least 40 occupied seats.
Consider a row of chairs with occupied seats such that each empty seat is next to an occupied seat. Then there are at most 2 empty seats between any of the n-1 pairs of consecutive occupied seats, at most 1 empty seat before the first occupied seat and at most 1 empty seat after the last occupied seat. Therefore, the row must have at most n + 2(n-1) + 2 = 3n chairs in total.
Therefore, a row with 39 or fewer occupied seats may have at most seats, which is too few. A row with 40 occupied seats and 120 total seats can be created by placing 1 empty seat on each end and 2 empty seats between each pair of consecutive occupied seats.
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