PUMaC 2008 · 几何（B 组） · 第 9 题

PUMaC 2008 — Geometry (Division B) — Problem 9

(7 points) Let H be the region of points ( x, y ), such that (1 , 0), ( x, y ), ( − x, y ), and ( − 1 , 0) form an isosceles trapezoid whose legs are shorter than the base between ( x, y ) and ( − x, y ). Find the least possible positive slope that a line could have without intersecting H .

解析

Infinitesimal Randall Munroe is glued to the center of a pentagon with side length 1. At each corner of the pentagon is a confused infinitesimal velociraptor. At any time, each raptor is running at one unit per second directly towards the next raptor in the pentagon (in counterclockwise order). How far does each confused raptor travel before it reaches Randall Munroe? √ 5 ( ANS: 1 + 5 The raptors remain arranged in a pentagon, so at any time, the distance between two adjacent ( ) ( √ ) 2 π 1 raptors is decreasing at a rate of 1 − cos = 5 − 5 , since a raptor is moving towards the 5 4 ( ) 2 π next one at a speed of 1, and the next one is moving away at a rate of cos . So, it takes a 5 √ 4 5 √ time of = 1 + seconds for them to be at distance 0 from each other, which is when they 5 5 − 5 √ 5 reach Randall. Since they travel at one unit per second, each raptor covers a distance of 1 + . 5 CB: IAF,AM)