CTK Insights

Here’s a lively problem that was offered at the 1975 USA Math Olympiad (Porblem 4): Circle C(E), with center E, and C(F) with center F, meet in points P and Q. A on C(E) and B on C(F) are such that AB passes through P. Find the position of A and B for which AP*PB is maximum. The curious thing is that there are a trigonometric solution and a synthetic solution that give two different constructions, but I do not see any relation between the two.

The problem led to several discussions:

1. Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
2. Problem 4, 1975 USA Math Olympiad: Normals and Tangents
3. Two Circles and One More
4. Problem 4, 1975 USA Math Olympiad and the Radical Axis

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1. Thébault’s Problem and Upgrades