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 20:01, 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 |