PUMaC 2020 · 几何（B 组） · 第 1 题

PUMaC 2020 — Geometry (Division B) — Problem 1

You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let θ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is b 100 × θ c ?

解析

You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let θ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is b 100 × θ c ? Proposed by: Henry Erdman Answer: 200 Let the radius of the turn be r and the width of the road w . Then, for a turn of angle θ , the outside path has length ( r + w ) θ . The inside path has length 2 w + rθ . These are equal when θ = 2, so our answer is 200.