HMMT 十一月 2013 · THM 赛 · 第 2 题

HMMT November 2013 — THM Round — Problem 2

[ 4 ] You are standing at a pole and a snail is moving directly away from the pole at 1 cm/s. When the snail is 1 meter away, you start “Round 1”. In Round n ( n ≥ 1), you move directly toward the snail at n + 1 cm/s. When you reach the snail, you immediately turn around and move back to the starting pole 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?

解析

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