Problem:
A disk of radius 1 rolls all the way around the inside of a square of side length and sweeps out a region of area . A second disk of radius 1 rolls all the way around the outside of the same square and sweeps out a region of area . The value of can be written as , where , and are positive integers and and are relatively prime. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
To obtain the region swept out by the first disk, remove a square of side length from the center of the original square and replace the 4 unit squares at the corners of the original square with 4 quarter-circles of radius 1 . The area of this region is
The region swept out by the second disk is the disjoint union of 4 rectangles, each with length and width 2 , and 4 quarter-circles of radius 2 , so its area is . Therefore
Solving this equation gives , so the requested sum is .
.jpg)
The problems on this page are the property of the MAA's American Mathematics Competitions