2003 AIME II Problems/Problem 4
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Problem
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Embed the tetrahedron in 4-space (It makes the calculations easier)
It's vertices are
, 
, 
, 
To get the center of any face, we take the average of the three coordinates of that face. The vertices of the center of the faces are:
,
,
,
The side length of the large tetrahedron is 
by the distance formula
The side length of the smaller tetrahedron is 
by the distance formula
Their ratio is 
, so the ratio of their volumes is 
See also
| 2003 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
