2001 AIME II Problems/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 1. 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
On the first construction, , four new tetrahedra will be constructed with side lengths of the original one. Since the ratio of the volume of similar polygons is the cube of the ratio of their corresponding lengths, it follows that each of these new tetrahedra will have volume . The total volume added here is then .
We now note that for each midpoint triangle we construct in step , there are now places to construct new midpoint triangles for step . The outward tetrahedron for the midpoint triangle provides of the faces, while the three equilateral triangles surrounding the midpoint triangle provide the other . This is because if you read this question carefully, it asks to add new tetrahedra to each face of which also includes the ones that were left over when we did the previous addition of tetrahedra. However, the volume of the tetrahedra being constructed decrease by a factor of . Thus we have the recursion , and so .
The volume of , and . Note that the summation was in fact a geometric series.
See also
https://www.math.hmc.edu/funfacts/ffiles/30002.2.shtml http://users.math.yale.edu/public_html/People/frame/Fractals/Labs/KochTetra/KochTetraAns3.html
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