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Find the answer for n = 1991 and n = 2000

Source: Vietnam TST 1991 for the 32nd IMO, problem 1

June 25, 2005
geometryparallelogramcircumcirclegeometry unsolved

Problem Statement

1.) In the plane let us consider a set SS consisting of n≥3n \geq 3 distinct points satisfying the following three conditions: I. The distance between any two points ∈S\in S is not greater than 1. II. For every point A∈SA \in S, there are exactly two “neighbor” points, i.e. two points X,Y∈SX, Y \in S for which AX=AY=1AX = AY = 1. III. For arbitrary two points A,B∈SA, B \in S, let A′,A′′A', A'' be the two neighbors of A,B′,B′′A, B', B'' the two neighbors of BB, then A′AA′′=B′BB′′A'AA'' = B'BB''. Is there such a set SS if n=1991n = 1991? If n=2000n = 2000 ? Explain your answer.
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