Problem of the Week #75: Monday December 9th, 2024
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 1970 Canadian Math Olympiad Q5.
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
a). If a triangle has 1 vertex on each side of an equilateral triangle of side 1, what’s its smallest possible perimeter?
b). If an n-sided polygon has 1 vertex on each side of a regular n-gon of side 1, what’s its smallest possible perimeter?
c). If a tetrahedron has one vertex on each face of a regular tetrahedron of edge length 1, what’s its smallest possible total edge length?
d). If a tetrahedron has one vertex on each face of a regular tetrahedron of edge length 1, what’s its smallest possible surface area?
e). If an octahedron has one vertex on each face of a cube of edge length 1, what’s its smallest possible total edge length?
f). If an octahedron has one vertex on each face of a cube of edge length 1, what’s its smallest possible surface area?
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.