HMMT 十一月 2013 · GEN 赛 · 第 2 题

HMMT November 2013 — GEN Round — Problem 2

[ 3 ] Plot points A, B, C at coordinates (0 , 0), (0 , 1), and (1 , 1) in the plane, respectively. Let S denote the union of the two line segments AB and BC . Let X be the area swept out when Bobby rotates 1 S counterclockwise 45 degrees about point A . Let X be the area swept out when Calvin rotates S 2 X + X 1 2 clockwise 45 degrees about point A . Find . 2

解析

[ 3 ] Plot points A, B, C at coordinates (0 , 0), (0 , 1), and (1 , 1) in the plane, respectively. Let S denote the union of the two line segments AB and BC . Let X be the area swept out when Bobby rotates 1 S counterclockwise 45 degrees about point A . Let X be the area swept out when Calvin rotates S 2 X + X 1 2 clockwise 45 degrees about point A . Find . 2 π Answer: It’s easy to see X = X . Simple cutting and pasting shows that X equals the area 1 2 1 4 √ √ X + X 1 1 2 π 1 2 of of a circle with radius AC = 2, so = X = π ( 2) = . 1 8 2 8 4