HMMT 二月 2006 · 冲刺赛 · 第 4 题

HMMT February 2006 — Guts Round — Problem 4

[6] Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of 10 km/hr. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at 20 km/hr; when going in the opposite direction the fly flies at 30 km/hr (thanks to a breeze). How many meters does the fly fly?

解析

Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of 10 km/hr. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at 20 km/hr; when going in the opposite direction the fly flies at 30 km/hr (thanks to a breeze). How many meters does the fly fly? 1 Answer: 55 Solution: Suppose that at a given instant the fly is at Ann and the two cars are 12 d apart. Then, while each of the cars travels 4 d , the fly travels 8 d and meets Anne. Then the fly turns around, and while each of the cars travels d , the fly travels 3 d and meets Ann again. So, in this process described, each car travels a total of 5 d while the 11 fly travels 11 d . So the fly will travel times the distance traveled by each bumper 5 11 50 car: · = 55 meters. 5 2