2024 AIME I Problems/Problem 14
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Problem
Let be a tetrahedron such that , , and . There exists a point inside the tetrahedron such that the distances from to each of the faces of the tetrahedron are all equal. This distance can be written in the form , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Solution 1
Notice that \(41=4^2+5^2\), \(89=5^2+8^2\), and \(80=8^2+4^2\), let \(A~(0,0,0)\), \(B~(4,5,0)\), \(C~(0,5,8)\), and \(D~(4,0,8)\). Then the plane \(BCD\) has a normal \begin{equation*} \mathbf n:=\frac14\overrightarrow{BC}\times\overrightarrow{CD}=\frac14\begin{pmatrix}-4\\0\\8\end{pmatrix}\times\begin{pmatrix}4\\-5\\0\end{pmatrix}=\begin{pmatrix}10\\8\\5\end{pmatrix}. \end{equation*} Hence, the distance from \(A\) to plane \(BCD\), or the height of the tetrahedron, is \begin{equation*} h:=\frac{\mathbf n\cdot\overrightarrow{AB}}{|\mathbf n|}=\frac{10\times4+8\times5+5\times0}{\sqrt{10^2+8^2+5^2}}=\frac{80\sqrt{21}}{63}. \end{equation*} Each side of the tetrahedron has the same area due to congruency by "S-S-S", and we call it \(S\). Then by the volume formula for cones, \begin{align*} \frac13Sh&=V_{D\text-ABC}=V_{I\text-ABC}+V_{I\text-BCD}+V_{I\text-CDA}+V_{I\text-DAB}\\ &=\frac13Sr\cdot4. \end{align*} Hence, \(r=\tfrac h4=\tfrac{20\sqrt{21}}{63}\), and so the answer is \(20+21+63=\boxed{104}\).
Solution by Quantum-Phantom
Solution 2
Inscribe tetrahedron in an rectangular prism as shown above.
By the Pythagorean theorem, we note
Solving yields and
Since each face of the tetrahedron is congruent, we know the point we seek is the center of the circumsphere of We know all rectangular prisms can be inscribed in a circumsphere, therefore the circumsphere of the rectangular prism is also the circumsphere of
We know that the distance from all faces must be the same, so we only need to find the distance from the center to plane .
Let and We obtain that the plane of can be marked as or and the center of the prism is
Using the Point-to-Plane distance formula, our distance is
Our answer is
Solution 3(Formula Abuse)
We use the formula for the volume of iscoceles tetrahedron.
Note that all faces have equal area due to equal side lengths. By Law of Cosines, we find .
From this, we find and can find the area of as
Let be the distance we want to find. By taking the sum of (equal) volumes We have Plugging in and simplifying, we get for an answer of
~AtharvNaphade
Solution 4
Let be perpendicular to that meets this plane at point . Let , , and be heights to lines , , and with feet , , and , respectively.
We notice that all faces are congruent. Following from Heron's formula, the area of each face, denoted as , is .
Hence, by using this area, we can compute , and . We have , , and .
Because , we have . Recall that . Hence, . Hence, .
Analogously, and .
We introduce a function for that is equal to 1 (resp. -1) if point and the opposite vertex of side are on the same side (resp. opposite sides) of side .
The area of is \begin{align*} A & = \epsilon_{BC} {\rm Area} \ \triangle HBC + \epsilon_{CD} {\rm Area} \ \triangle HCD + \epsilon_{BD} {\rm Area} \ \triangle HBD \\ & = \frac{1}{2} \epsilon_{BC} BC \cdot HP + \frac{1}{2} \epsilon_{CD} CD \cdot HQ + \frac{1}{2} \epsilon_{BD} BD \cdot HR \\ & = \frac{1}{2} \epsilon_{BC} BC \cdot \sqrt{AP^2 - AH^2} + \frac{1}{2} \epsilon_{CD} CD \cdot \sqrt{AQ^2 - AH^2} \\ & \quad + \frac{1}{2} \epsilon_{BD} CD \cdot \sqrt{AR^2 - AH^2} . \hspace{1cm} (1) \end{align*}
Denote . The above equation can be organized as \begin{align*} B & = \epsilon_{BC} \sqrt{B^2 - 89 AH^2} + \epsilon_{CD} \sqrt{B^2 - 41 AH^2} \\ & \quad + \epsilon_{BD} \sqrt{B^2 - 80 AH^2} . \end{align*}
This can be further reorganized as \begin{align*} B - \epsilon_{BC} \sqrt{B^2 - 89 AH^2} & = \epsilon_{CD} \sqrt{B^2 - 41 AH^2} + \epsilon_{BD} \sqrt{B^2 - 80 AH^2} . \end{align*}
Taking squares on both sides and reorganizing terms, we get \begin{align*} & 16 AH^2 - \epsilon_{BC} B \sqrt{B^2 - 89 AH^2} \\ & = \epsilon_{CD} \epsilon_{BD} \sqrt{\left( B^2 - 41 AH^2 \right) \left( B^2 - 80 AH^2 \right)} . \end{align*}
Taking squares on both sides and reorganizing terms, we get
Taking squares on both sides, we finally get \begin{align*} AH & = \frac{20B}{189} \\ & = \frac{40A}{189}. \end{align*}
Now, we plug this solution to Equation (1). We can see that , . This indicates that is out of . To be specific, and are on opposite sides of , and are on the same side of , and and are on the same side of .
Now, we compute the volume of the tetrahedron , denoted as . We have .
Denote by the inradius of the inscribed sphere in . Denote by the incenter. Thus, the volume of can be alternatively calculated as \begin{align*} V & = {\rm Vol} \ IABC + {\rm Vol} \ IACD + {\rm Vol} \ IABD + {\rm Vol} \ IBCD \\ & = \frac{1}{3} r \cdot 4A . \end{align*}
From our two methods to compute the volume of and equating them, we get \begin{align*} r & = \frac{10A}{189} \\ & = \frac{20 \sqrt{21}}{63} . \end{align*}
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 5(A quicker method to compute the height from to plane )
We put the solid to a 3-d coordinate system. Let , . We put on the plane. Now ,we compute the coordinates of .
Applying the law of cosines on , we get . Thus, . Thus, .
Denote with .
Because , we have \[ x^2 + y^2 + z^2 = 89 \hspace{1cm} (1) \]
Because , we have \[ \left( x - \sqrt{80} \right)^2 + y^2 + z^2 = 41 \hspace{1cm} (2) \]
Because , we have \[ \left( x - \frac{4}{\sqrt{5}} \right)^2 + \left( y - \frac{3 \sqrt{21}}{\sqrt{5}} \right)^2 + z^2 = 80 \hspace{1cm} (3) \]
Now, we compute , and .
Taking , we get \[ 2 \sqrt{80} x = 128 . \]
Thus, .
Taking , we get \[ 2 \cdot \frac{4}{\sqrt{5}} x + 2 \cdot \frac{3 \sqrt{21}}{\sqrt{5}} y = 50 . \]
Thus, .
Plugging and into Equation (1), we get .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution 1 by OmegaLearn.org
Video Solution 2
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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