IMO 1975 SL 15

Origin: USS

Problem

Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?

Solution

Assume that the center of the circle is at the origin O(0, 0), and that the points A1, A2, . . . , A1975 are arranged on the upper half-circle so that \angleAiOA1 = \alphai (\alpha1 = 0). The distance AiAj equals 2 sin \alphaj−\alphai

2 sin \alphaj 2 cos \alphai 2 −cos \alphaj 2 sin \alphai 2 , and it will be rational if all sin \alphak 2 , cos \alphak are rational. Finally, observe that there exist infinitely many angles \alpha such that both sin \alpha, cos \alpha are rational, and that such \alpha can be arbitrarily small. For example, take \alpha so that sin \alpha = 2t t2+1 and cos \alpha = t2−1 t2+1 for any t \inQ.