Mock Geometry AIME 2011 Problems/Problem 9
is a right pyramid with square base edge length 6, and The probability that a randomly selected point inside the pyramid is at least units away from each face can be expressed in the form where are relatively prime positive integers. Find
[I believe solution is wrong - IP' is not sqrt(6)/3 as stated in the second paragraph - The_Turtle]
Let be the set of all points that are at least units away from each face. is tetrahedron, and it is similar to . This can be proved by showing that is bounded by 5 planes, each of which is parallel to a corresponding plane of . Let the vertices of be such that is the closest vertex to and so forth. Consider cross section . This cross section contains two concentric, similar triangles, and . Furthermore, these triangles are equilateral; is the diagonal of a square with a side length of and so .
From symmetry it follows that . Let intersect at and at . Then . We can calculate , it is the height of an equilateral triangle with a side length of . Then . Similarly, let be the sidelenth of . Then is the height of this triangle and so is equal to . Let be the foot of the perpendicular from to . bisects by symmetry, and so and . Also as it just the distance from to .
Plugging these values in yields . Solving yields . Therefore the ratio to is . The ratio of their volumes is then the ratio of their sides cubed, or . The ratio of the volumes of to is equivalent to the probability a point will be in . Hence and .