Problem of the Week #73: Monday November 25th, 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 YouTube video.
In the square below, points that are closer to the center than one of the vertices is coloured yellow. The area of the yellow region is 1/2 of the area of the square.
a). In an equilateral triangle, what fraction of its area is closer to the center than one of the vertices?
b). Find the fractional area closer to the center than one of the vertices in a regular n-gon.
c). Point P is halfway between the center and the left side of the square. What fraction of the square’s area is closer to point P than one of the vertices?
d). Can we calculate the fractional area closer to any point P in a square than one of the vertices?
e). Can we calculate the fractional area closer to any point P in a regular n-gon than one of the vertices?
f). What fraction of the volume of a regular tetrahedron is closer to its center than one of the vertices?
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.