Question 97
Let $A$ be the set of points in the plane whose coordinates are integers and let
$F$ be the collection of all functions from $A$ to $\{1, -1\}$. We call a function $f$ in $F$ perfect if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition
$$
\sum_{0 \lt d(P,Q) \lt 2010} \frac{f(P)f(Q) - g(P)g(Q)}{d(P,Q)} \geq 0
$$
where $d(P, Q)$ denotes the distance between $P$ and $Q$. Show that there exist infinitely many perfect functions that are not translates of each other.
$F$ be the collection of all functions from $A$ to $\{1, -1\}$. We call a function $f$ in $F$ perfect if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition
$$
\sum_{0 \lt d(P,Q) \lt 2010} \frac{f(P)f(Q) - g(P)g(Q)}{d(P,Q)} \geq 0
$$
where $d(P, Q)$ denotes the distance between $P$ and $Q$. Show that there exist infinitely many perfect functions that are not translates of each other.