Math Gold Medalist

Lor

2023 AIME I

Problem 13

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.

[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]

Volume

Pythagorean Theorem

Suppose 2 Faces that have common edge are ABCD and ABEF

Find AB

Suppose AC = sqrt(31) and BD = sqrt(21)

We should find ratio of their heights

Calculate height of each of them by applying Pythagorean theorem

case1) AE = sqrt(31)  and  BF = sqrt(21)

case1) AE = sqrt(21)  and  BF = sqrt(31)

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   Solution