Difference between revisions of "2001 AIME II Problems/Problem 12"
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== Problem == | == Problem == | ||
| + | Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra <math>P_{i}</math> is defined recursively as follows: <math>P_{0}</math> is a regular tetrahedron whose volume is 1. To obtain <math>P_{i + 1}</math>, replace the midpoint triangle of every face of <math>P_{i}</math> by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of <math>P_{3}</math> is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
== Solution == | == Solution == | ||
| + | {{solution}} | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2001|n=II|num-b=11|num-a=13}} | |
Revision as of 00:44, 20 November 2007
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
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See also
| 2001 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
