ELMO 2014

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

Let $n$ be a positive integer and let $a_1$ , $a_2$ , \dots, $a_n$ be real numbers strictly between $0$ and $1$ . For any subset $S$ of $\{1, 2, ..., n\}$ , define $$f(S) = \prod_{i \in S} a_i \cdot \prod_{j \not \in S} (1-a_j).$$ Suppose that $\sum_{|S| \text{ odd}} f(S) = \frac{1}{2}$ . Prove that $a_k = \frac{1}{2}$ for some $k$ . (Here the sum ranges over all subsets of $\{1, 2, ..., n\}$ with an odd number of elements.)

Proposed by Kevin Sun

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ . Let $\omega_1$ and $\omega_2$ denote the circumcircles of triangles $BOC$ and $BHC$ , respectively. Suppose the circle with diameter $\overline{AO}$ intersects $\omega_1$ again at $M$ , and line $AM$ intersects $\omega_1$ again at $X$ . Similarly, suppose the circle with diameter $\overline{AH}$ intersects $\omega_2$ again at $N$ , and line $AN$ intersects $\omega_2$ again at $Y$ . Prove that lines $MN$ and $XY$ are parallel.

Proposed by Sammy Luo