Question 1
In the plane, let $S$ consist of $n\ge3$ distinct points satisfying: (1) the distance between any two points of $S$ is at most 1; (2) each point $A$ has exactly two neighbors, meaning points of $S$ at distance 1 from $A$; (3) if $A',A''$ are the neighbors of $A$ and $B',B''$ those of $B$, then $\angle A' AA''=\angle B' BB''$ for any $A,B\in S$. Does such a set exist for $n=1991$? For $n=2000$? Explain.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.