MEMO 2013

Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $AB = 2PQ$ , $BC = 2QR$ , $CA = 2 RP$ , and the lines $PQ, QR,$ and $RP$ pass through the points $A, B ,$ and $C$ , respectively. (All six points $A, B, C, P, Q,$ and $R$ are distinct.)

Let $K$ be a point inside an acute triangle $ABC$ , such that $BC$ is a common tangent of the circumcircles of $AKB$ and $AKC$ . Let $D$ be the intersection of the lines $CK$ and $AB$ , and let $E$ be the intersection of the lines $BK$ and $AC$ . Let $F$ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$ . The circumcircle of $ABC$ and the circle $k$ with centre $F$ and radius $FD$ intersect at points $P$ and $Q$ . Prove that the segment $PQ$ is a diameter of $k$ .

The numbers from 1 to $2013^2$ are written row by row into a table consisting of $2013 \times 2013$ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $1, 4, 9, \cdots, 2013^2$ are simultaneously deleted. How many cells remain?

The expression $$\pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box$$ is written on the blackboard. Two players, $A$ and $B$ , play a game, taking turns. Player $A$ takes the first turn. In each turn, the player on turn replaces a symbol $\Box$ by a positive integer. After all the symbols $\Box$ are replace, player $A$ replaces each of the signs $\pm$ by either + or -, independently of each other. Player $A$ wins if the value of the expression on the blackboard is not divisible by any of the numbers $11, 12, \cdots, 18$ . Otherwise, player $B$ wins. Determine which player has a winning strategy.