Problem:
A right rectangular prism (i.e., a rectangular parallelepiped) 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?
Solution:
Let with be the sides of , the rectangular parallelepiped that is similar to . Since is cut from by a plane parallel to one of the faces of , two of the numbers must equal two of the numbers . Furthermore
so it follows that and . Thus
so . Now has factors, and follows by substituting and into . Thus the triple can be selected in ways.
Each of these choices for and results in a rectangular parallelepiped of the type desired. Indeed, if and , then cutting by a plane parallel to the face and at distance from that face produces an parallelepiped similar to .
The problems on this page are the property of the MAA's American Mathematics Competitions