HMMT 十一月 2013 · GEN 赛 · 第 2 题

英文解析

2. [ 3 ] Plot points A, B, C at coordinates (0 , 0), (0 , 1), and (1 , 1) in the plane, respectively. Let S denotethe union of the two line segments AB and BC . Let X be the area swept out when Bobby rotates
   S counterclockwise 45 degrees about point A . Let X be the area swept out when Calvin rotates S1
   X + X2
   1 2
   clockwise 45 degrees about point A . Find .
   π2
   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) = .
   8 2 8 41