IMO 1971 SL 2

Origin: BUL

Problem

Prove that for every natural number m \geq1 there exists a finite set Sm of points in the plane satisfying the following condition: If A is any point in Sm, then there are exactly m points in Sm whose distance to A equals 1.

Solution

We will construct such a set Sm of 2m points. Take vectors u1, . . . , um in a given plane, such that |ui| = 1/2 and 0 ̸= |c1u1 + c2u2 + \cdot \cdot \cdot + cnun| ̸= 1/2 for any choice of numbers ci equal to 0 or \pm1. Such vectors are easily constructed by induction on m: For u1, . . . , um−1 fixed, there are only finitely many vector values um that vi- olate the upper condition, and we may set um to be any other vector of length 1/2. Let Sm be the set of all points M0 + \epsilon1u1 + \epsilon2u2 + \cdot \cdot \cdot + \epsilonmum, where M0 is any fixed point in the plane and \epsiloni = \pm1 for i = 1, . . . , m. Then Sm obviously satisfies the condition of the problem.