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Difference between revisions of "2023 AIME I Problems/Problem 13"
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==Problem 13==
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==Problem==
 
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths <math>\sqrt{21}</math> and <math>\sqrt{31}</math>.
 
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths <math>\sqrt{21}</math> and <math>\sqrt{31}</math>.
 
The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math>
 
The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math>

Revision as of 00:21, 10 February 2023

Problem

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.

{insert diagram here}

Solution 1 (3-D Vector Analysis)

Denote $\alpha = \tan^{-1} \frac{\sqrt{21}}{\sqrt{31}}$. Denote by $d$ 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 $x-O-y$ plane. For this bottom face, we put a vertex with an acute angle $2 \alpha$ at the origin, denoted as $O$. For two edges that are on the bottom face and meet at $O$, we put one edge on the positive side of the $x$-axis. The endpoint is denoted as $A$. Hence, $A = \left( d , 0 , 0 \right)$. We put the other edge in the first quadrant of the $x-O-y$ plane. The endpoint is denoted as $B$. Hence, $B = \left( d \cos 2 \alpha , d \sin 2 \alpha , 0 \right)$.

For the third edge that has one endpoint $O$, we denote by $C$ its second endpoint. We denote $C = \left( u , v , w \right)$. Without loss of generality, we set $w > 0$. Hence, \[ u^2 + v^2 + w^2 = d^2 . \hspace{1cm} (1) \]

We have \begin{align*} \cos \angle AOC & = \frac{\overrightarrow{OA} \cdot \overrightarrow{OC}}{|OA| \cdot |OC|} \\ & = \frac{u}{d} , \hspace{1cm} (2) \end{align*} and \begin{align*} \cos \angle BOC & = \frac{\overrightarrow{OB} \cdot \overrightarrow{OC}}{|OB| \cdot |OC|} \\ & = \frac{u \cos 2 \alpha + v \sin 2 \alpha}{d} . \hspace{1cm} (3) \end{align*}

Case 1: $\angle AOC = \angle BOC = 2 \alpha$ or $2 \left( 90^\circ - \alpha \right)$.

By solving (2) and (3), we get \begin{align*} u & = \pm d \cos 2 \alpha , \\ v & = \pm d \cos 2 \alpha \frac{1 - \cos 2 \alpha}{\sin 2 \alpha} \\ & = \pm d \cos 2 \alpha \tan \alpha . \end{align*}

Plugging these into (1), we get \begin{align*} w & = d \sqrt{1 - \cos^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha} \\ & = d \sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha} . \hspace{1cm} (4) \end{align*}

Case 2: $\angle AOC = 2 \alpha$ and $\angle BOC = 2 \left( 90^\circ - \alpha \right)$, or $\angle BOC = 2 \alpha$ and $\angle AOC = 2 \left( 90^\circ - \alpha \right)$.

By solving (2) and (3), we get \begin{align*} u & = \pm d \cos 2 \alpha , \\ v & = \mp d \cos 2 \alpha \frac{1 + \cos 2 \alpha}{\sin 2 \alpha} \\ & = \mp d \cos 2 \alpha \cot \alpha . \end{align*}

Plugging these into (1), we get \begin{align*} w & = d \sqrt{1 - \cos^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha} \\ & = d \sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha} . \hspace{1cm} (5) \end{align*}

We notice that $(4) > (5)$. 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 \begin{align*} \frac{(4)}{(5)} & = \frac{\sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha}} {\sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha}} \\ & = \sqrt{\frac{\tan^2 2 \alpha - \tan^2 \alpha}{\tan^2 2 \alpha - \cot^2 \alpha}} . \end{align*}

Recall that $\tan \alpha = \frac{\sqrt{21}}{\sqrt{31}}$. Thus, $\tan 2 \alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} = \frac{\sqrt{21 \cdot 31}}{5}$. Plugging this into the equation above, we get \begin{align*} \frac{(4)}{(5)} & = \frac{63}{62}. \end{align*}

Therefore, the answer is $63 + 62 = \boxed{\textbf{(125) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 2 (no trig)

Let one of the vertices be at the origin and the three adjacent vertices be $u$, $v$, and $w$. For one of the parallelepipeds, the three diagonals involving the origin have length $\sqrt {21}$. Hence, $(u+v)\cdot (u+v)=u\cdot u+v\cdot v+2u\cdot v=21$ and $(u-v)\cdot (u-v)=u\cdot u+v\cdot v-2u\cdot v=31$. Since all of $u$, $v$, and $w$ have equal length, $u\cdot u=13$, $v\cdot v=13$, and $u\cdot v=-2.5$. Symmetrically, $w\cdot w=13$, $u\cdot w=-2.5$, and $v\cdot w=-2.5$. Hence the volume of the parallelepiped is given by $\sqrt{\operatorname{det}\begin{pmatrix}13&-2.5&-2.5\\-2.5&13&-2.5\\-2.5&-2.5&13\end{pmatrix}}=\sqrt{\operatorname{det}\begin{pmatrix}15.5&-15.5&0\\-2.5&13&-2.5\\0&-15.5&15.5\end{pmatrix}}=\sqrt{15.5^2\operatorname\det\begin{pmatrix}1&-1&0\\-2.5&13&-2.5\\0&-1&1\end{pmatrix}}=\sqrt{15.5^2\cdot 8}$.

For the other parallelpiped, the three diagonals involving the origin are of length $\sqrt{31}$ and the volume is $\sqrt{\operatorname{det}\begin{pmatrix}13&2.5&2.5\\2.5&13&2.5\\2.5&2.5&13\end{pmatrix}}=\sqrt{\operatorname{det}\begin{pmatrix}10.5&-10.5&0\\2.5&13&2.5\\0&-10.5&10.5\end{pmatrix}}=\sqrt{10.5^2\operatorname\det\begin{pmatrix}1&-1&0\\2.5&13&2.5\\0&-1&1\end{pmatrix}}=\sqrt{10.5^2\cdot 18}$.

Consequently, the answer is $\sqrt\frac{10.5^2\cdot 18}{15.5^2\cdot 8}=\frac{63}{62}$, giving $\boxed{125}$.

~EVIN-

2023 AIME I (Problems • Answer Key • Resources)
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