AIME 2023 I · 第 13 题

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

$$\begin{aligned} \cos \angle AOC & = \frac{\overrightarrow{OA} \cdot \overrightarrow{OC}}{|OA| \cdot |OC|} \\ & = \frac{u}{d} , \hspace{1cm} (2) \end{aligned}$$ and

$$\begin{aligned} \cos \angle BOC & = \frac{\overrightarrow{OB} \cdot \overrightarrow{OC}}{|OB| \cdot |OC|} \\ & = \frac{u \cos 2 \alpha + v \sin 2 \alpha}{d} . \hspace{1cm} (3) \end{aligned}$$ Case 1: $\angle AOC = \angle BOC = 2 \alpha$ or $2 \left( 90^\circ - \alpha \right)$.

By solving (2) and (3), we get

$$\begin{aligned} u & = \pm d \cos 2 \alpha , \\ v & = \pm d \cos 2 \alpha \frac{1 - \cos 2 \alpha}{\sin 2 \alpha} \\ & = \pm d \cos 2 \alpha \tan \alpha . \end{aligned}$$ Plugging these into (1), we get

$$\begin{aligned} w & = d \sqrt{1 - \cos^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha} \\ & = d \sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha} . \hspace{1cm} (4) \end{aligned}$$ 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

$$\begin{aligned} u & = \pm d \cos 2 \alpha , \\ v & = \mp d \cos 2 \alpha \frac{1 + \cos 2 \alpha}{\sin 2 \alpha} \\ & = \mp d \cos 2 \alpha \cot \alpha . \end{aligned}$$ Plugging these into (1), we get

$$\begin{aligned} w & = d \sqrt{1 - \cos^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha} \\ & = d \sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha} . \hspace{1cm} (5) \end{aligned}$$ 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

$$\begin{aligned} \frac{(4)}{(5)} & = \frac{\sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \tan^2 \alpha}} {\sqrt{\sin^2 2 \alpha - \cos^2 2 \alpha \cot^2 \alpha}} \\ & = \sqrt{\frac{\tan^2 2 \alpha - \tan^2 \alpha}{\tan^2 2 \alpha - \cot^2 \alpha}} . \end{aligned}$$ 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

$$\begin{aligned} \frac{(4)}{(5)} & = \frac{63}{62}. \end{aligned}$$ 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 parallelepiped, 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-

Note on the linear algebra formula: It is well known that a parallelepiped whose 3 adjacent vertices have vectors $u= \langle u_1, u_2, u_3 \rangle$, $v= \langle v_1, v_2, v_3 \rangle$, and $w = \langle w_1, w_2, w_3 \rangle$ volume is the absolute value of $\operatorname{det} \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix}$ Then the volume is also

\begin{align*} \sqrt{\left (\operatorname{det} \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{pmatrix}\right)^2} &= \sqrt{\operatorname{det} \left( \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{pmatrix} \cdot \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{pmatrix} \right) } \\ &= \sqrt{\operatorname{det} \left( \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{pmatrix} \cdot \begin{pmatrix} u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3 \\ \end{pmatrix} \right) } = \sqrt{\operatorname{det} \begin{pmatrix} u \cdot u & u \cdot v & u \cdot w \\ v \cdot u & v \cdot v & v \cdot w \\ w \cdot u & w \cdot v & w \cdot w \end{pmatrix} } \end{align*}

Since the determinant of the transpose of a matrix equals to the determinant of the original matrix.

Solution 3 (No trig, no linear algebra)

Observe that both parallelepipeds have two vertices (one on each base) that have three congruent angles meeting at them. Denote the parallelepiped with three acute angles meeting $P$, and the one with three obtuse angles meeting $P'$.

The volume of a parallelepiped is simply the base area times the height, but because both parallelepipeds have the same base, what we want is just the ratio of the heights.

Denote the point with three acute angles meeting at it in $P$ as $A$, and its neighbors $B$, $C$, and $D$. Similarly, denote the point with three obtuse angles meeting at it in $P'$ as $A'$, and its neighbors $B'$, $C'$, and $D'$.

We have the following equations:

$$\textrm{Height of }P\textrm{ from }ACD = \frac{\textrm{Vol}(ABCD) \cdot 3}{[ACD]},$$ $$\textrm{Height of }P'\textrm{ from }A'C'D' = \frac{\textrm{Vol}(A'B'C'D') \cdot 3}{[A'C'D']}.$$ However, $ACD$ and $A'C'D'$ are both half the area of a rhombus with diagonals $\sqrt{31}$ and $\sqrt{21}$, so our ratio is really

$$\frac{P}{P'} = \frac{\textrm{Vol}(ABCD)}{\textrm{Vol}(A'B'C'D')}.$$ Because the diagonals of all of the faces are $\sqrt{31}$ and $\sqrt{21}$, each edge of the parallelepipeds is $\sqrt{13}$ by the Pythagorean theorem.

We have $AB = AC = AD = \sqrt{13}$, and $BC = CD = BD = \sqrt{21}$. When we drop a perpendicular to the centroid of $BCD$ from $A$ (let's call this point $O$), we have $BO = \frac{\sqrt{21}}{\sqrt{3}} = \sqrt{7}$. Thus,

$$AB^2 - BO^2 = AO^2$$ $$13 - 7 = AO^2 = 6$$ $$AO = \sqrt{6}.$$ The area of base $BCD$ is $\frac{21\sqrt{3}}{4}$. Hence,

$$\textrm{Vol}(ABCD) = \frac{\sqrt{6}\cdot\frac{21\sqrt{3}}{4}}{3}$$ $$= \frac{63\sqrt{2}}{12}.$$ We can apply a similar approach to $A'B'C'D'$.

$A'B' = A'C' = A'D' = \sqrt{13}$, and $B'C' = C'D' = B'D' = \sqrt{31}$. When we drop a perpendicular to the centroid of $B'C'D'$ from $A'$ (let's call this point $O'$), we have $B'O' = \frac{\sqrt{31}}{\sqrt{3}} = \sqrt{\frac{31}{3}}$. Thus,

$$A'B'^2 - B'O'^2 = A'O'^2$$ $$13 - \frac{31}{3} = A'O'^2$$ $$A'O' = \sqrt{\frac{8}{3}} = \frac{2\sqrt{6}}{3}.$$ The area of base $B'C'D'$ is $\frac{31\sqrt{3}}{4}$. Hence,

$$\textrm{Vol}(A'B'C'D') = \frac{\frac{2\sqrt{6}}{3}\cdot\frac{31\sqrt{3}}{4}}{3}$$ $$= \frac{186\sqrt{2}}{36}$$ $$= \frac{62\sqrt{2}}{12}.$$ Finally,

$$\frac{P}{P'} = \frac{\textrm{Vol}(ABCD)}{\textrm{Vol}(A'B'C'D')} = \frac{\frac{63\sqrt{2}}{12}}{\frac{62\sqrt{2}}{12}} = \frac{63}{62}.$$ Our answer is $63 + 62 = \boxed{125}$.

~mathboy100

Solution 4 (Pythagorean theorem)

Since the two parallelepipeds have the same base, all we need to do is to find their respective heights.

As illustrated in the above diagram, drop a perpendicular from $D$ onto the base at $P$. Denote the center of the base by $O$. By symmetry, $P$ must be on $AO$. Now we need to find $DP$.

Apply Pythagorean theorem to $\triangle DPA$ we have

$$DP^2 = AD^2 - AP^2.$$ Apply Pythagorean theorem to $\triangle DPC$ and then $\triangle CPO$ we have

$$DP^2 = DC^2 - CP^2 = DC^2 - (CO^2 + OP^2) = DC^2 - (CO^2 + (AO-AP)^2) = DC^2 - CO^2 - (AO-AP)^2.$$ Combining the above two, we have

$$AD^2 - AP^2 = DC^2 - CO^2 - (AO-AP)^2.$$ Since $AD=\sqrt{13}$, $DC=\sqrt{21}$, $CO=\frac{\sqrt{21}}{2}$, $AO=\frac{\sqrt{31}}{2}$, plug them into the above equation and solve for the only unknown variable $AP$, we get $AP=\frac{5}{\sqrt{31}}.$

Thus the height

$$DP = \sqrt{AD^2 - AP^2} = \sqrt{13 - \frac{25}{31}} = \sqrt{\frac{378}{31}}.$$

For the other parallelepiped, using the same approach and drop a perpendicular from $E'$ onto the base at $P'$. Similarly applying Pythagorean theorem to $\triangle E'P'C'$, $\triangle E'P'A'$ and $\triangle A'P'O'$ we have

$$C'E'^2 - C'P'^2 = A'E'^2 - A'O'^2 - (C'P'-C'O')^2.$$ Plugging known values into the above equation and solve for the only unknown variable $C'P'$, we get $C'P'=\frac{5}{\sqrt{21}}.$

Thus the height

$$E'P' = \sqrt{C'E'^2 - C'P'^2} = \sqrt{13 - \frac{25}{21}} = \sqrt{\frac{248}{21}}.$$ The ratio between the two is therefore

$$\frac{DP}{E'P'} = \frac{\sqrt{\frac{378}{31}}}{\sqrt{\frac{248}{21}}} = \sqrt{\frac{2\cdot3^3\cdot7}{31}\cdot\frac{3\cdot7}{2^3\cdot31}} = \frac{3^2\cdot7}{2\cdot31} = \frac{63}{62}$$ , giving $\boxed{125}$.

~sgdzw

Solution 5 (Visual)

Let us inscribe a tetrahedron $ACB'D'$ in given parallelepiped so that its edges coincide with the diagonals of the faces of the parallelepiped. Note that the three edges outgoing from the vertex $B'$ have the same length $b$, and the three edges at the base have a different length $a.$

The volume of the tetrahedron $V= \frac {a^2}{12}\sqrt{3b^2 - a^2}$ is three times less than the volume of the parallelepiped.

In second parallelepiped $a$ and $b$ change the positions.

Required ratio is $\frac {a^2 \cdot \sqrt {3b^2 - a^2}}{b^2 \cdot \sqrt {3a^2 - b^2}} = \frac {21 \sqrt {3 \cdot 31 - 21}}{31 \cdot \sqrt{3 \cdot 21 - 31}} = \frac {21 \sqrt {72}}{31 \sqrt {32}}= \frac {63}{62}.$

Claim

Let $ABCD$ be the regular pyramid, $AB = AC = AD = b, BC = BD = CD = a.$

The area of $\triangle BCD = \frac {a^2 \sqrt {3}}{4}.$

Height $AO^2 = AB^2 - OB^2 = {b^2 - \frac {a^2}{3}}.$

Volume $V= \frac {a^2}{12}\sqrt{3b^2 - a^2}.$

vladimir.shelomovskii@gmail.com, vvsss

Solution 6

Let $\angle CAB$ be acute and let $X$ be the foot of the altitude from $C$ to $AB$. Notice that this also implies that $X$ is the foot of the altitude from $D$ to $AB$. Now $\sqrt{13} \cdot CX = AB \cdot CX = \frac{1}{2} \sqrt{21} \cdot \sqrt{31}$ so $CX = \frac{\sqrt{21} \cdot \sqrt{31}}{2 \sqrt{13}}$ and $DX$ is the same. $CD$ must either be $\sqrt{21}$ or $\sqrt{31}$ because it is a diagonal. If $CD = \sqrt{31}$, applying the Law of Cosines on $\angle DXC$, $\cos \angle DXC = -\frac{5}{21}$ so $\sin \angle DXC = \sqrt{\frac{416}{441}}$. If $CD = \sqrt{21}$, $\cos \angle DXC = \frac{5}{31}$, so $\sin \angle DXC = \sqrt{\frac{936}{961}}$. The ratios of the two parallelepipeds is equal to the ratios of the heights (since the bases are the same) which is equal to the ratio of the sines. Therefore it is

$$\frac{\sqrt{\frac{936}{961}}}{\sqrt{\frac{416}{441}}} = \sqrt{\frac{936}{416}} \cdot \frac{21}{31} = \frac{3}{2} \cdot \frac{21}{31} = \frac{63}{62}$$ so the answer is $63 + 62 = \boxed{125}$.

Video Solution

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Animated Video Solution

https://youtu.be/VvCl5KIqT9M

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