1999 AMC 12 Problem 29
Below is the professionally curated solution for Problem 29 of the 1999 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.
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Difficulty rating: 2380
29.
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point is selected at random inside the circumscribed sphere. The probability that lies inside one of the five small spheres is closest to
Solution:
Let be the common center of the inscribed and circumscribed spheres. Splitting the tetrahedron into four congruent pieces from shows the circumradius is times the inradius, so the circumscribed sphere has times the inscribed sphere's volume
Each externally tangent sphere fits between a face and the circumscribed sphere and is congruent to the inscribed sphere, so the five small spheres have total volume The probability is closest to
Thus, the correct answer is C.