2003 AIME II Problems/Problem 4

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

Solution 1

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

Solution 2

Let the large tetrahedron be , and the small tetrahedron be , with on , on , on , and on . Clearly, the two regular tetrahedrons are similar, so if we can find the ratio of the sides, we can find the ratio of the volumes. Let , for our convenience. Dropping an altitude from to , and calling the foot , we have . Since . By Law of Cosines, we have . Hence, the ratio of the volumes is .