Problem of the Week #70: Monday November 4th, 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 AIME 1984 #6.
All straight lines that bisect the area of a circle go through its center.
a). Find all straight lines bisecting the area covered by two identical circles in both the overlapping and the non-overlapping cases.
b). Find all straight lines bisecting the area covered by two partially overlapping circles of different sizes.
c). If we draw all bisecting lines of these two circles and find all points where at least two of the lines intersect, what shape do these points make up?
d). Find all straight lines bisecting the total area of these partially overlapping circles of different sizes if the area of each circle on either side of the line is calculated separately (ie the overlapping region counts as twice the non-overlapping region of the same area).
e). Find all straight lines bisecting the area covered by these three identical and symmetrically overlapping circles.
f). Find all straight lines bisecting the area covered by these three identical overlapping circles.
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.