Sets in the Plane

A. Does there exist a set A in the plane such that its intersection with every circle contains exactly two points?

B. Does there exist a set B in the plane such that its intersection with every circle of radius 1 contains exactly two points?

Difficulty level (1 very easy - 10 very hard): 4

Topics:
Geometry -> Plane Geometry -> Circles Proof and Example -> Constructing an Example / Counterexample Set Theory Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems
Sources:
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