HMMT 二月 2006 · CALC 赛 · 第 3 题

HMMT February 2006 — CALC Round — Problem 3

At time 0, an ant is at (1 , 0) and a spider is at ( − 1 , 0). The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the x -axis. It so happens that the ant’s horizontal speed is always half the spider’s. What will the shortest distance ever between the ant and the spider be? ∞ 4 ∑ k

解析

At time 0, an ant is at (1 , 0) and a spider is at ( − 1 , 0). The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the x -axis. It so happens that the ant’s horizontal speed is always half the spider’s. What will the shortest distance ever between the ant and the spider be? √ 14 Answer: 4 Solution: Picture an instant in time where the ant and spider have x -coordinates a and s , respectively. If 1 ≤ s ≤ 3, then a ≤ 0, and the distance between the bugs is at least 1. If s > 3, then, needless to say the distance between the bugs is at least 2. If − 1 ≤ s ≤ 1, then s = 1 − 2 a , and the distance between the bugs is √ 2 √ √ (8 a − 3) + 7 2 2 2 ( a − (1 − 2 a )) + (1 − a ) = 8 a − 6 a + 2 = , 8 √ which attains the minimum value of 7 / 8 when a = 3 / 8. ∞ 4 ∑ k