HMMT 二月 2006 · 冲刺赛 · 第 1 题

英文解析

1. A bear walks one mile south, one mile east, and one mile north, only to find itselfwhere it started. Another bear, more energetic than the first, walks two miles south,
   two miles east, and two miles north, only to find itself where it started. However, thebears are not white and did not start at the north pole. At most how many milesapart, to the nearest . 001 mile, are the two bears’ starting points?
   Answer: 3 . 477
   Solution: Say the first bear walks a mile south, an integer n > 0 times aroundthe south pole, and then a mile north. The middle leg of the first bear’s journey isa circle of circumference 1 /n around the south pole, and therefore about miles 1
   2 nπnorth of the south pole. (This is not exact even if we assume the Earth is perfectlyspherical, but it is correct to about a micron.) Adding this to the mile that the bearwalked south/north, we find that it started about 1 + miles from the south pole.1
   2 nπ
   Similarly, the second bear started about 2 + miles from the south pole for some 2
   2 mπinteger m > 0, so they must have started at most
   1 2 3
   3 + + ≤ 3 + ≈ 3 . 477
   2 nπ 2 mπ 2 πmiles apart.