Problem:
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures by by units. Given that the volume of this set is , where , and are positive integers, and and are relatively prime, find .
Solution:
First consider the points in the six parallelepipeds projecting unit outward from the original parallelepiped. Two of these six parallelepipeds are by by , two are by by , and two are by by . The sum of their volumes is . Next consider the points in the twelve quartercylinders of radius whose heights are the edges of the original parallelepiped. The sum of their volumes is . Finally, consider the points in the eight octants of a sphere of radius at the eight vertices of the original parallelepiped. The sum of their volumes is . Because the volume of the original parallelepiped is , the requested volume is , so .
The problems on this page are the property of the MAA's American Mathematics Competitions