1995 AIME Problem 11

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

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.