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2021 MPFG Problem 19

Revision as of 03:34, 27 August 2025 by Cassphe (talk | contribs) (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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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.

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