2005 AIME II Problems/Problem 10
Let the side of the octahedron be of length . Let the vertices of the octahedron be so that and are opposite each other and . The height of the square pyramid is and so it has volume and the whole octahedron has volume .
Let be the midpoint of , be the midpoint of , be the centroid of and be the centroid of . Then and the symmetry ratio is (because the medians of a triangle are trisected by the centroid), so . is also a diagonal of the cube, so the cube has side-length and volume . The ratio of the volumes is then and so the answer is .
Let the octahedron have vertices . Then the vertices of the cube lie at the centroids of the faces, which have coordinates . The cube has volume 8. The region of the octahedron lying in each octant is a tetrahedron with three edges mutually perpendicular and of length 3. Thus the octahedron has volume , so the ratio is and so the answer is .
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