Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Right Rectangular Prism.
A right rectangular prism P (that is rectangular parallelopiped) 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 has non-zero volume, given that b=1995, find number of ordered tuples (a,b,c) does such a plane exist.
Integers
Divisibility
Algebra
Answer: is 40.
AIME I, 1995, Question 11
Geometry Vol I to IV by Hall and Stevens
Let Q be similar to P
Let sides of Q be x,y,z for \(x \leq y \leq z\)
then \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c} < 1\)
As one face of Q is face of P
or, P and Q has at least two side lengths in common
or, x <a, y<b, z<c
or, y=a, z=b=1995
or, \(\frac{x}{a}=\frac{a}{1995}=\frac{1995}{c}\)
or, \(ac=1995^{2}=(3)^{2}(5)^{2}(7)^{2}(19)^{2}\)
or, number of factors of \((3)^{2}(5)^{2}(7)^{2}(19)^{2}\)=(2+1)(2+1)(2+1)(2+1)=81
or, \([\frac{81}{2}]=40\) for a <c.
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