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 1 of 5: Choose good angles
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In plain words

A chord of the unit circle has length 2sin⁡θ2\sin\theta when θ\theta is the inscribed half-angle. Thus rational sine values produce rational chord lengths.

sin⁡θ∈Q,cos⁡θ∈Q\sin\theta\in\mathbb Q,\quad \cos\theta\in\mathbb Q
A rational-angle chord on the unit circle
Unit circle showing a chord determined by an angle whose sine and cosine are rational
Detailed analysis

Every primitive Pythagorean triple gives a rational point (cos⁡θ,sin⁡θ)(\cos\theta,\sin\theta) on the unit circle, so infinitely many angles have both sine and cosine rational. Call them good.