Difference between revisions of "2021 MPFG Problem 19"
m (→Solution 1) |
(→Solution 1) |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 4: | Line 4: | ||
==Solution 1== | ==Solution 1== | ||
| − | Connect O 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>. | + | 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>. |
<math>S</math> equals to <math>1</math> regular tetrahedron with <math>4</math> protruding tetrahedrons. | <math>S</math> equals to <math>1</math> regular tetrahedron with <math>4</math> protruding tetrahedrons. | ||
| − | [[File: | + | [[File:New3d.png|600px|center]] |
| + | |||
| + | [[File:2d.png|400px|]] [[File:Protrudes.png|500px|]] | ||
<math>S_{tetra} = (\frac{5}{3})^3 = \frac{125}{27}</math> | <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 | ~cassphe | ||
Latest revision as of 05:38, 27 August 2025
Problem
Let 
be a regular tetrahedron. Let 
be the regular tetrahedron whose vertices are the centers of the faces of . Let 
be the circumcenter of either tetrahedron. Given a point 
different from , let 
be the midpoint of the points of intersection of the ray 
with and . Let 
be the set of eight points m(P) where P is a vertex of either or . What is the volume of the convex hull of divided by the volume of ? Express your
answer as a fraction in simplest form.
Solution 1
Connect with the 4 vertices of . Extend the line made by connecting the top vertex of with , intersecting at the base/vertex of .
equals to 
regular tetrahedron with 
protruding tetrahedrons.
~cassphe
