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functional geometry, d(a,b)=n->d(f(a),f(b))=n for n=1 implies ∀n∈N

Source: Mongolia 1999 Teachers secondary level P6

May 6, 2021
fefunctional equationfunctional geometrygeometry

Problem Statement

Let ff be a map of the plane into itself with the property that if d(A,B)=1d(A,B)=1, then d(f(A),f(B))=1d(f(A),f(B))=1, where d(X,Y)d(X,Y) denotes the distance between points XX and YY. Prove that for any positive integer nn, d(A,B)=nd(A,B)=n implies d(f(A),f(B))=nd(f(A),f(B))=n.
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