AMC 8 1995 Problem 25
AMC8 1995 年第 25 题
行程问题★★★★☆
Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?
从 Dallas 到 Houston 的巴士每小时正点发车。从 Houston 到 Dallas 的巴士每小时半点发车。从一城到另一城的行程需 5 小时。假设巴士在同一条高速公路上行驶，一辆从 Houston 出发的巴士在途中（不在车站）会与多少辆 Dallas 方向的巴士相遇？
- A.5
- B.6
- C.9
- D.10
- E.11
正确答案 · Correct Answer
D
解析 · Solution
解法1
常规解法 · 算重叠时间窗
设你乘的休斯顿方向车 12:30 出发、17:30 到达，路上 5 小时。一辆整点 发的达拉斯方向车在路上的时段为 到 h+5。你们会在路上相遇当且仅当两段相交且相向交错，即 。整点 取 8 到 17 共 10 个，故遇到 10 辆。
巧解 · 每半小时遇一辆
对向车每小时一班；两车相向、合速度是单车的两倍，所以你每半小时就与一辆对向车相遇。5 小时里共遇 辆。
Solution 1
Elementary · the overlap time window
Suppose your Houston-bound bus leaves at 12:30 and arrives at 17:30 (5 hours). A Dallas-bound bus leaving on the hour is on the road from to h+5. You meet it on the highway exactly when the intervals overlap with the two of you approaching, i.e. . The on-the-hour values are from 8 to 17, which is 10 buses.
Slick · meet one every half hour
Opposing buses leave once an hour, but since both buses move toward each other the closing speed doubles, so you meet an oncoming bus every half hour. Over your 5-hour trip that is buses.
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