Team Selection Test for IMO 2010

Let $L$ be a line in the plane, and let ${\pi }_1$ and ${\pi }_2$ be the corresponding open half-planes. We set $$f_L(P)=\{\begin{array}{ll}1& \text{if }P\in {\pi }_1\cup L,\\ -1& \text{if }P\in {\pi }_2.\end{array}$$ We will show that $f_L$ is a perfect function. Then the family $\{f_L:(0,0)\in L\}\subset \mathcal{F}$ consists of infinitely many perfect functions that are not translates of each other.

Let $g$ be a function in $\mathcal{F}$ differing from $f_L$ at finitely many points, and let $$K_{\pm }=\{(P,Q):f_L(P)f_L(Q)-g(P)g(Q)=\pm 2\text{ and }0<d(P,Q)<2010\}.$$ Then $$\sum _{0<d(P,Q)<2010}\frac{f_L(P)f_L(Q)-g(P)g(Q)}{d(P,Q)}=\sum _{(P,Q)\in K_{+}}\frac{2}{d(P,Q)}-\sum _{(P,Q)\in K_{-}}\frac{2}{d(P,Q)}$$

We will prove that this expression is nonnegative by defining an injection $m:K_{-}\to K_{+}$ that satisfies $d(m(P,Q))=d(P,Q)$.

Let $(P,Q)\in K_{-}$. Then $f_L(P)\ne f_L(Q)$ and $g(P)=g(Q)$. Consider the line $\ell$ passing through the points $P$ and $Q$. Let $P_0=P$, $P_1=Q$, and let $P_i$ be the unique point on $\ell$ such that $d(P_i,P_{i-1})=d(P,Q)$ and $P_i\ne P_{i-2}$ for $i\ge 2$, and $d(P_i,P_{i+1})=d(P,Q)$ and $P_i\ne P_{i+2}$ for $i\le -1$.

Since $f_L$ and $g$ differ at finitely many points, we know that $f_L(P_i)=g(P_i)$ for all $i$ with sufficiently large $|i|$. In particular, $g$ changes sign finitely many times, but at least once on $\ell$. Let $k$ be the smallest integer such that $g(P_k)\ne g(P_{k+1})$. We define $m(P,Q)=(P_k,P_{k+1})$.

$(P_k,P_{k+1})\in K_{+}$ as $f_L(P_k)f_L(P_{k+1})-g(P_k)g(P_{k+1})=1-(-1)=2$, and we have $d(P_k,P_{k+1})=d(P,Q)$ by construction. Finally, since $(P,Q)$ is the only pair among $(P_i,P_{i+1})$, $i\in \mathbb{Z}$, satisfying $f_L(P_i)\ne f_L(P_{i+1})$, $m$ is injective.