Problem of the Week #88: Monday April 28th, 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 1975 USAMO Q4.
Two circles intersect at P and Q. Let’s call line AB through P a combined chord if AP and PB are chords of the respective circles. Let’s define the split product of AB as the length of AP times the length of PB.
a). Draw a different combined chord through P that has the same split product as AB.
b). Which of the following combined chords has a greater split product?
c). Draw a different combined chord through P that has the same split product as AB.
d). Find all combined chords through P that have a split product of 0.
e). Find all combined chords through P that do not share a split product with a different combined chord.
f). Find the combined chord through P that has the greatest split product.
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.