HMMT 二月 2012 · 冲刺赛 · 第 17 题

HMMT February 2012 — Guts Round — Problem 17

[ 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 directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed 1 (William’s reverses direction at the same speed). Initially, Mark’s orb is moving at meters/s, and 1000 William’s orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens? 1 2 2 3 3

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 directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed 1 (William’s reverses direction at the same speed). Initially, Mark’s orb is moving at meters/s, and 1000 William’s orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens? 17 Answer: 2 / 125 If the two orbs leave their respective walls at the same time, then they will return to their walls at the same time (because colliding affects both their speeds). After returning to the n 4 n 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