Problem of the Week #79: Monday February 3rd, 2025
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by the 1973 USAMO Q1.
Let’s consider the 3D version of regular planar angles, called solid angles. If an angle can be thought of as a portion of a full circle, then a solid angle is a portion of a full sphere.
a). An angle that takes up a full circle is 2π units (ie. radians). By the same token, how many units should we make a solid angle that takes up a full sphere?
b). Using the same unit we developed in the previous question, how big is the interior solid angle at a vertex of a cube?
c). What fraction of a sphere does a vertex of a regular tetrahedron take up? (Or roughly how many regular tetrahedrons can we fit around a point?)
d). What fraction of a sphere does a vertex of a regular octahedron take up? (Or roughly how many regular octahedrons can we fit around a point?)
e). All triangles have the same interior angle sum of π units (radians). Do all tetrahedrons have the same interior solid angle sum?
f). What’s the maximum possible the interior solid angle sum of a tetrahedron? What’s the minimum?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.