HMMT 二月 2006 · CALC 赛 · 第 3 题

英文解析

3. At time 0, an ant is at (1 , 0) and a spider is at ( − 1 , 0). The ant starts walkingcounterclockwise along the unit circle, and the spider starts creeping to the right alongthe 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?
   √
   Answer:14
   Solution: Picture an instant in time where the ant and spider have x -coordinates a 4
   and s , respectively. If 1 ≤ s ≤ 3, then a ≤ 0, and the distance between the bugs is atleast 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