Question 11
A right rectangular prism $P$ (i.e., a rectangular parallelepiped) has sides of integral length $a,$ $b,$ $c,$ with $a\leq b\leq c.$ 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?