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Difference between revisions of "2021 MPFG Problem 19"

(Created page with "==Problem== Let <math>T</math> be a regular tetrahedron. Let <math>t</math> be the regular tetrahedron whose vertices are the centers of the faces of <math>T</math>. Let <math...")
 
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Let <math>T</math> be a regular tetrahedron. Let <math>t</math> be the regular tetrahedron whose vertices are the centers of the faces of <math>T</math>. Let <math>O</math> be the circumcenter of either tetrahedron. Given a point <math>P</math> different from <math>O</math>, let <math>m(P)</math> be the midpoint of the points of intersection of the ray <math>\overrightarrow{OP}</math> with <math>t</math> and <math>T</math>. Let <math>S</math> be the set of eight points m(P) where P is a vertex of either <math>t</math> or <math>T</math>. What is the volume of the convex hull of <math>S</math> divided by the volume of <math>t</math>? Express your
 
Let <math>T</math> be a regular tetrahedron. Let <math>t</math> be the regular tetrahedron whose vertices are the centers of the faces of <math>T</math>. Let <math>O</math> be the circumcenter of either tetrahedron. Given a point <math>P</math> different from <math>O</math>, let <math>m(P)</math> be the midpoint of the points of intersection of the ray <math>\overrightarrow{OP}</math> with <math>t</math> and <math>T</math>. Let <math>S</math> be the set of eight points m(P) where P is a vertex of either <math>t</math> or <math>T</math>. What is the volume of the convex hull of <math>S</math> divided by the volume of <math>t</math>? Express your
 
answer as a fraction in simplest form.
 
answer as a fraction in simplest form.
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==Solution 1==
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Connect <math>O</math> with the 4 vertices of <math>T</math>. Extend the line made by connecting the top vertex of <math>T</math> with <math>O</math>, intersecting at the base/vertex of <math>t</math>.
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<math>S</math> equals to <math>1</math> regular tetrahedron with <math>4</math> protruding tetrahedrons.
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[[File:New3d.png|600px|center]]
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[[File:2d.png|400px|]] [[File:Protrudes.png|500px|]]
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<math>S_{tetra} = (\frac{5}{3})^3 = \frac{125}{27}</math>
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<math>S_{total} = \frac{125}{27} \cdot (1+\frac{\frac{4}{3}}{\frac{5}{3}}) = \boxed{\frac{25}{3}}</math>
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~cassphe

Latest revision as of 05:38, 27 August 2025

Problem

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points m(P) where P is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$? Express your answer as a fraction in simplest form.

Solution 1

Connect $O$ with the 4 vertices of $T$. Extend the line made by connecting the top vertex of $T$ with $O$, intersecting at the base/vertex of $t$.

$S$ equals to $1$ regular tetrahedron with $4$ protruding tetrahedrons.

New3d.png

2d.png Protrudes.png

$S_{tetra} = (\frac{5}{3})^3 = \frac{125}{27}$

$S_{total} = \frac{125}{27} \cdot (1+\frac{\frac{4}{3}}{\frac{5}{3}}) = \boxed{\frac{25}{3}}$

~cassphe

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