HMMT 二月 2001 · CALC 赛 · 第 1 题
HMMT February 2001 — CALC Round — Problem 1
题目详情
英文原题
- A sequence of ants walk from (0 , 0) to (1 , 0) in the plane. The n th ant walks alongn semicircles of radius with diameters lying along the line from (0 , 0) to (1 , 0). Let L be 1
the length of the path walked by the n th ant. Compute lim L .nnnn →∞
5 3
解析
英文解析
- A sequence of ants walk from (0 , 0) to (1 , 0) in the plane. The n th ant walks alongn semicircles of radius with diameters lying along the line from (0 , 0) to (1 , 0). Let L be 1
the length of the path walked by the n th ant. Compute lim L .nnnn →∞
( )
1 1 2 π
Solution: A semicircle of radius has length π = , so n such semicircles haven 2 n ntotal length π .
5 3