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Turkey TST 2010 Q6

Source:

September 1, 2010
analytic geometryfunctioncombinatorics proposedcombinatorics

Problem Statement

Let Λ\Lambda be the set of points in the plane whose coordinates are integers and let FF be the collection of all functions from Λ\Lambda to {1,−1}.\{1,-1\}. We call a function ff in FF perfect if every function gg in FF that differs from ff at finitely many points satisfies the condition ∑0<d(P,Q)<2010f(P)f(Q)−g(P)g(Q)d(P,Q)≥0 \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 where d(P,Q)d(P,Q) denotes the distance between PP and Q.Q. Show that there exist infinitely many perfect functions that are not translates of each other.
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