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AIME 1995 Problem 11

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A right rectangular prism PP_{} (i.e., a rectangular parallelpiped) has sides of integral length a,b,c,a, b, c, with a≤b≤c.a\le b\le c. A plane parallel to one of the faces of PP_{} cuts PP_{} into two prisms, one of which is similar to P,P_{}, and both of which have nonzero volume. Given that b=1995,b=1995, for how many ordered triples (a,b,c)(a, b, c) does such a plane exist?

Dataset text AIME 1983 to 2026. Quoted with source.

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