¶ 2001 AIME II Problem 12
Problem:
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra is defined recursively as follows: is a regular tetrahedron whose volume is . To obtain , replace the midpoint triangle of every face of by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of is , where and are relatively prime positive integers. Find .
Solution:
The diagram shows . Notice that has triangular faces, has , and, inductively, has . This expression therefore counts the small tetrahedra that are attached to to form . The volume of each of these small tetrahedra is , and hence the volume of is more than the volume of . In particular, the volume of is
Thus .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions