HMMT 十一月 2013 · THM 赛 · 第 2 题

英文解析

2. [ 4 ] You are standing at a pole and a snail is moving directly away from the pole at 1 cm/s. When thesnail is 1 meter away, you start “Round 1”. In Round n ( n ≥ 1), you move directly toward the snailat n + 1 cm/s. When you reach the snail, you immediately turn around and move back to the startingpole at n + 1 cm/s. When you reach the pole, you immediately turn around and Round n + 1 begins.
   At the start of Round 100, how many meters away is the snail?
   Answer: 5050 Suppose the snail is x meters away at the start of round n , so x = 1, and the runnern 1
   100 x 100 xn ntakes = seconds to catch up to the snail. But the runner takes the same amount of time
   ( n +1) − 1 n
   200 x 1 2 xn nto run back to the start, so during round n , the snail moves a distance of x − x = = .
   n +1 nn 100 n
   101! / 2!
   101 101 100
   Finally, we have x = x = x = · · · = x = 5050.
   100 99 98 1
   99 99 98 99!