Problem of the Week #69: Monday October 28th, 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.
Please register for an account if you would like to join the discussion below or share your own problems.
This problem is inspired by AIME 1983 #15.
A circle of radius 5 has chord AB with length 6. A second chord is drawn here and is bisected by AB.
a). Find a different chord that is bisected by AB.
b). Sketch the area made up of all chords on the circle that are bisected by AB.
c). Calculate the area made up of all chords of the circle of radius 5 that are bisected by AB if AB is 6 units long.
d). Find a formula for the area made up of all chords of the circle that are bisected by AB in terms of the length of AB.
e). A sphere of radius 5 has a circular cross section with diameter 6. Sketch the shape made up of all lines between two points on the sphere that are bisected by the cross section.
f). A sphere of radius 5 has a circular cross section with diameter 6. Find the volume of the shape made up of all lines between two points on the sphere that are bisected by the cross section.
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.