2003 AIME I Problems/Problem 5
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
The set can be broken into several parts: the big 
parallelepiped, 
external parallelepipeds that each share a face with the large parallelepiped and have a height of 
, the 
spheres (one centered at each vertex of the large parallelepiped), and the 
cylinders connecting each adjacent pair of spheres.
- The volume of the parallelepiped is
cubic units. - The volume of the external parallelepipeds is
. - There are
of the 
spheres, each of radius 
. Together, their volume is 
. - There are
of the 
cylinders, so 
complete cylinders can be formed. Their volumes are 
, 
, and 
, adding up to 
.
cubic units.
.
of the 
.
of the 
, 
, and 
, adding up to 
.The combined volume of these parts is 
. Thus, the answer is 
.
See also
| 2003 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
