PUMaC 2014 · 团队赛 · 第 2 题

PUMaC 2014 — Team Round — Problem 2

[ 4 ] Given a Pacman of radius 1, and mouth opening angle 90 , what is the largest (circular) pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the p radius be equal to a b + c where b is square free. Find a + b + c . 2 x 6

解析

[ 4 ] Given a Pacman of radius 1, and mouth opening angle 90 , what is the largest (circular) pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the √ circumcircle of the Pacman. Let the radius be equal to a b + c where b is square free. Find a + b + c . Solution: Inscribe a smaller circle within the circle sector and draw the line from the center of the larger circle through the center of the inscribed circle to the edge of the circle. Then the length of this line segment is 1. Dropping perpendiculars from the center of the smaller circle, we get 1 √ a square and thus the length is also r + r 2 where r is the radius of the smaller circle. So √ √ r (1 + 2) = 1 ⇒ r = 2 − 1 ⇒ a + b + c = 2 . 2 x − 6