Problem:
What is the volume of the region in three-dimensional space defined by the inequalities and ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
In the first octant, the first inequality reduces to , and the inequality defines the region under a plane that intersects the coordinate
axes at , and . By symmetry, the first inequality defines the region inside a regular octahedron centered at the origin and having internal diagonals of length 2. The upper half of this octahedron is a pyramid with altitude 1 and a square base of side length , so the volume of the octahedron is . The second inequality defines the region obtained by translating the first region up 1 unit. The intersection of the two regions is bounded by another regular octahedron with internal diagonals of length 1 . Because the linear dimensions of the third octahedron are half those of the first, its volume is that of the first, or .
The problems on this page are the property of the MAA's American Mathematics Competitions