OIM 2013

A set $S$ of positive integers is said to be channeler if for any three distinct numbers $a,b,c \in S$ , we have $a\mid bc$ , $b\mid ca$ , $c\mid ab$ .

a) Prove that for any finite set of positive integers $\{ c_1, c_2, \ldots, c_n \}$ there exist infinitely many positive integers $k$ , such that the set $\{ kc_1, kc_2, \ldots, kc_n \}$ is a channeler set.

b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.

Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$ . Let $A$ and $B$ be two points on the segment $XY$ . The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$ , respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$ . Let $M$ the the intersection point between $XY$ and $NP$ . Prove that $M$ is the midpoint of the segment $AB$ .

Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$ . The point $D \ne E$ in on the segment $OE$ . The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$ . The lines $EB$ and $CD$ meets at point $F$ . The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$ .

Prove that the line $EI$ is the angle bissector of $\angle BEC$ .

Let $A$ and $B$ be two sets such that $A \cup B$ is the set of the positive integers, and $A \cap B$ is the empty set. It is known that if two positive integers have a prime larger than $2013$ as their difference, then one of them is in $A$ and the other is in $B$ . Find all the possibilities for the sets $A$ and $B$ .

A beautiful configuration of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$ .