Difference between revisions of "2021 MPFG Problem 19"

Revision as of 05:04, 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.

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

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

@@ Line 8: / Line 8: @@
 <math>S</math> equals to <math>1</math> regular tetrahedron with <math>4</math> protruding tetrahedrons.
-[[File:FDG.png|250px|center]] [[File:FDG.png|250px|center]]
+[[File:3d.png|250px|center]] [[File:2d.png|250px|center]]
 <math>S_{tetra} = (\frac{5}{3})^3 = \frac{125}{27}</math>
 <math>S_{total} = \frac{125}{27} \cdot (1+\frac{4}{5}) = \frac{25}{3}</math>

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Difference between revisions of "2021 MPFG Problem 19"

Revision as of 05:04, 27 August 2025

Problem

Solution 1