2023 AIME I Problems/Problem 13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths and
.
The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is
, where
and
are relatively prime positive integers. Find
. A parallelepiped is a solid with six parallelogram faces
such as the one shown below.
Solution 1 (3-d Vector Analysis)
Denote .
Denote by
the length of each side of a rhombus.
Now, we put the solid to the 3-d coordinate space.
We put the bottom face on the plane.
For this bottom face, we put a vertex with an acute angle
at the origin, denoted as
.
For two edges that are on the bottom face and meet at
, we put one edge on the positive side of the
-axis. The endpoint is denoted as
. Hence,
.
We put the other edge in the first quadrant of the
plane. The endpoint is denoted as
. Hence,
.
For the third edge that has one endpoint , we denote by
its second endpoint.
We denote
.
Without loss of generality, we set
.
Hence,
We have
and
Case 1: or
.
By solving (2) and (3), we get
Plugging these into (1), we get
Case 2: and
, or
and
.
By solving (2) and (3), we get
Plugging these into (1), we get
We notice that . Thus, (4) (resp. (5)) is the parallelepiped with a larger (resp. smaller) height.
Therefore, the ratio of the volume of the larger parallelepiped to the smaller one is
Recall that .
Thus,
.
Plugging this into the equation above, we get
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)