Problem of the Week #56: Monday July 29th, 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 this twitter post.
For any point inside a square, the sum of the distances from this point to the four sides is half of the perimeter.
a). What’s the sum of the distances from a point inside an equilateral triangle to the sides?
b). Find a formula for the the sum of the distances from a point inside a regular n-gon to the sides.
c). What’s the sum of the perpendicular distances from a point outside a regular n-gon to the sides?
d). For any point inside a cube, the sum of distances from this point to the six faces is three times the side length. What’s the sum of the distances from a point inside a regular tetrahedron to the faces?
e). What’s the sum of the distances from a point inside an octahedron to the faces?
f). What’s the sum of the perpendicular distances from a point outside an octahedron to the faces?
g). What about the other two platonic solids?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.