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Function on a set of n concentric circles

Source: Romanian IMO Team Selection Test TST 1996, problem 15

September 27, 2005
functiongeometry proposedgeometry

Problem Statement

Let S S be a set of n n concentric circles in the plane. Prove that if a function f:S→S f: S\to S satisfies the property d(f(A),f(B))≥d(A,B) d( f(A),f(B)) \geq d(A,B) for all A,B∈S A,B \in S , then d(f(A),f(B))=d(A,B) d(f(A),f(B)) = d(A,B) , where d d is the euclidean distance function.
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