OMM 2017

A knight is placed on each square of the first column of a $2017 \times 2017$ board. A move consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$ -th column to contain one knight after a finite number of moves.

Note: Two squares are a knight-step away if they are opposite corners of a $2 \times 3$ or $3 \times 2$ board.

A set of $n$ positive integers is said to be balanced if for each integer $k$ with $1 \leq k \leq n$ , the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$ .

Let $ABC$ be an acute triangle with orthocenter $H$ . The circle through $B, H$ , and $C$ intersects lines $AB$ and $AC$ at $D$ and $E$ respectively, and segment $DE$ intersects $HB$ and $HC$ at $P$ and $Q$ respectively. Two points $X$ and $Y$ , both different from $A$ , are located on lines $AP$ and $AQ$ respectively such that $X, H, A, B$ are concyclic and $Y, H, A, C$ are concyclic. Show that lines $XY$ and $BC$ are parallel.

A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.

On a circle $\Gamma$ , points $A, B, N, C, D, M$ are chosen in a clockwise order in such a way that $N$ and $M$ are the midpoints of clockwise arcs $BC$ and $AD$ respectively. Let $P$ be the intersection of $AC$ and $BD$ , and let $Q$ be a point on line $MB$ such that $PQ$ is perpendicular to $MN$ . Point $R$ is chosen on segment $MC$ such that $QB = RC$ , prove that the midpoint of $QR$ lies on $AC$ .

Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$ , in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$ , there exists a box containing $n$ ballots. For each $n$ , find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.