ELMO SL 2013 G14

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$ . Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$ .

Now let $L$ be the set of lines containing exactly one point of $S(T)$ . Call a line $\ell_0$ passing through $O$ bad if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$ .

(a) Prove that $L$ is nonempty.

(b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$ , there exists a positive integer $n$ with $\ell(n) = \ell_0$ .

Proposed by David Yang