HMMT-Feb 2012 guts, Problem 17

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 (William's reverses direction at the same speed). Initially, Mark's orb is moving at $\frac{1}{1000}$ meters/s, and 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?

Solution: 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 wall $n$ times, Mark's orb will travel at $\frac{4^{n}}{1000}$ meter/s and William's will travel at $2^{n}$ meter/s. Mark wins when his orb is traveling faster at $n=10 \cdot \frac{4^{10}}{1000}=\frac{2^{17}}{125}$ $\fbox{2^{17} / 125}$.

Answer: 2^{17} / 125