1995 AIME Problems/Problem 11
A right rectangular prism (i.e., a rectangular parallelpiped) has sides of integral length with A plane parallel to one of the faces of cuts into two prisms, one of which is similar to and both of which have nonzero volume. Given that for how many ordered triples does such a plane exist?
Let be the prism similar to , and let the sides of be of length , such that . Then
Note that if the ratio of similarity was equal to , we would have a prism with zero volume. As one face of is a face of , it follows that and share at least two side lengths in common. Since , it follows that the only possibility is . Then,
The number of factors of is . Only in of these cases is (for , we end with a prism of zero volume). We can easily verify that these will yield nondegenerate prisms, so the answer is .
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