Difference between revisions of "2021 MPFG Problem 19"

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.

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

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

~cassphe

@@ Line 14: / Line 14: @@
 <math>S_{tetra} = (\frac{5}{3})^3 = \frac{125}{27}</math>
-<math>S_{total} = \frac{125}{27} \cdot (1+\frac{4}{5}) = \boxed{\frac{25}{3}}</math>
+<math>S_{total} = \frac{125}{27} \cdot (1+\frac{\frac{4}{3}}{\frac{5}{3}}) = \boxed{\frac{25}{3}}</math>
 ~cassphe

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Difference between revisions of "2021 MPFG Problem 19"

Latest revision as of 05:38, 27 August 2025

Problem

Solution 1