MathLabs

Problem 5

Determine, with proof, whether one can find 19751975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.
Step 4 of 5: Add one point while preserving all good angles
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∠BP=±∠AP±∠AB\angle BP=\pm\angle AP\pm\angle AB
Detailed analysis

Suppose finitely many points already have every pair-angle good. Choose a good angle avoiding the finitely many positions that would duplicate an old point, and place BB so that ∠AB\angle AB is this angle. For each old point PP, ∠BP\angle BP is a signed sum of ∠AP\angle AP and ∠AB\angle AB, hence is good by Step 2.