2001 AIME II Problem 12

Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra P_{i} is defined recursively as follows: P_{0} is a regular tetrahedron whose volume is 1. To obtain P_{i + 1}, replace the midpoint triangle of every face of P_{i} by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of P_{3} is \frac {m}{n}, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.