HMMT 二月 2012 · 冲刺赛 · 第 17 题
HMMT February 2012 — Guts Round — Problem 17
题目详情
英文原题
[11] Mark and William are playing a game. Two walls are placed 1 meter apart, w ith Mark and William each starting an orb at one of the walls. Simultaneously, they relea se their orbs directly toward the other. Both orbs are enchanted such that, upon colliding with each o ther, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted hi s orb so that when it collides with a wall it instantly reverses direction and goes at doubl e its previous speed (William’s reverses direction at the same speed). Initially, Mark’s orb is moving at1 1000meters/s, and William’s orb is moving at 1 meter/s. Mark wins when his orb passes the half way point between the two walls. How fast, in meters/s, is his orb going when this first happens?
解析
英文解析
- [ 11 ] Mark and William are playing a game. Two walls are placed 1 meter apart, with Mark and
William each starting an orb at one of the walls. Simultaneously, they release their orbs directlytoward the other. Both orbs are enchanted such that, upon colliding with each other, they instantlyreverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb sothat when it collides with a wall it instantly reverses direction and goes at double its previous speed
(William’s reverses direction at the same speed). Initially, Mark’s orb is moving at meters/s, and 1
1000
William’s orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between thetwo walls. How fast, in meters/s, is his orb going when this first happens?
Answer: 2 / 125 If the two orbs leave their respective walls at the same time, then they will return 17
to their walls at the same time (because colliding affects both their speeds). After returning to thenn 4
wall n times, Mark’s orb will travel at meter/s and William’s will travel at 2 meter/s. Mark
1000
10 17
4 2
wins when his orb is traveling faster at n = 10. =
1000 125
2 2 3 3 1