Problem of the Week #76: Monday December 16th, 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 Q8 of the 1970 Canadian Math Olympiad.
Consider all 1 unit long line segments that have an end point on the line y=0 and the other on x=0.
a). Sketch the shape traced by the midpoints of all these line segments.
b). Write the equation of the locus of midpoints of all 1 unit long line segments that have 1 end point on the line y=0 and the other on x=0.
c). Sketch the shape traced by the midpoints of all 1 unit long line segments that have 1 end point on the line y=0 and the other on y=x.
d). Write the equation of the locus of midpoints of all 1 unit long line segments that have 1 end point on the line y=0 and the other on y=x.
e). Sketch the shape traced by the midpoints of all 1 unit long line segments that have 1 end point on the line y=0 and the other on y=x/2.
f). Write the equation of the locus of midpoints of all 1 unit long line segments that have 1 end point on the line y=0 and the other on y=x/2.
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.