Problem of the Week #81: Monday February 17th, 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 Q3.
a). If we randomly select 3 vertices of a regular hexagon to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle?
b). If we randomly select 3 vertices of a regular polygon with 2n sides to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle?
c). If we randomly select 4 vertices of a regular octahedron to form a tetrahedron (If the 4 points are coplanar, let’s treat it as a flat tetrahedron.), what’s the probability that the center of the octahedron is on the surface of the tetrahedron?
d). If we randomly select 4 vertices of a cube to form a tetrahedron (if the 4 points are coplanar, let’s treat it as a flat tetrahedron.), what are the probabilities that the center of the cube is i) inside, ii) outside, iii) on the surface of the tetrahedron?
e). If we randomly select 4 vertices of a cube to form a tetrahedron, what’s the probability that the tetrahedron is flat (the 4 points are coplanar)?
f). If we randomly select 3 vertices of a cube to form a triangle, what are the probabilities that the center of the cube i) is inside, ii) is outside (but on the same plane as), iii) is on the edge of, iv) is not on the same plane as the triangle?
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.