ELMO 2014 Problem 3

We say a finite set $S$ of points in the plane is very if for every point $X$ in $S$ , there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).

(a) Fix an integer $n$ . Prove that if $n \ge 2$ , then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$ .

(b) Find the largest possible size of a very set not contained in any line.

(Here, an inversion with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$ .)

Proposed by Sammy Luo