Difference between revisions of "2003 AIME II Problems/Problem 4"
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== Solution == | == Solution == | ||
| − | {{ | + | Embed the tetrahedron in 4-space (It makes the calculations easier) |
| + | It's vertices are | ||
| + | <math>(1,0,0,0)</math>, <math>(0,1,0,0)</math>, <math>0,0,1,0)</math>, <math>(0,0,0,1)</math> | ||
| + | |||
| + | 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: | ||
| + | <math>(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 0)</math>,<math>(\frac{1}{3}, \frac{1}{3},0, \frac{1}{3})</math>,<math>(\frac{1}{3},0, \frac{1}{3}, \frac{1}{3})</math>,<math>(0,\frac{1}{3}, \frac{1}{3}, \frac{1}{3})</math> | ||
| + | |||
| + | The side length of the large tetrahedron is <math>\sqrt{2}</math> by the distance formula | ||
| + | The side length of the smaller tetrahedron is <math>\frac{\sqrt{2}}{3}</math> by the distance formula | ||
| + | |||
| + | Their ratio is <math>1:3</math>, so the ratio of their volumes is <math>\left(\frac{1}{3}\right)^3 = \frac{1}{27}</math> | ||
| + | |||
| + | <math>m+n = 1 + 27 = \boxed{028}</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2003|n=II|num-b=3|num-a=5}} | {{AIME box|year=2003|n=II|num-b=3|num-a=5}} | ||
Revision as of 21:01, 17 March 2008
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 | ||
