1975 IMO Problems/Problem 5
Determine, with proof, whether or not one can find points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.
Since there are infinitely many primitive Pythagorean triples, there are infinitely many angles s.t. are both rational. Call such angles good. By angle-sum formulas, if are good, then are also good.
For points on the circle , let be the angle subtended by . Now inductively construct points on s.t. all angles formed by them are good; for 1,2 take any good angle. If there are points chosen, pick a good angle and a marked point s.t. the point on with is distinct from the points. Since there are infinitely many good angles but finitely many marked points, such exists. For a previously marked point we have for suitable choices for the two . Since are both good, it follows that is good, which finishes induction by adding .
Observe that these points for work: since for on the circle, it follows that is rational, and so we're done.
The above solution was posted and copyrighted by tobash_co. The original thread for this problem can be found here: 
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