OMM 1999

On a table there are $1999$ counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types: (1) Remove several counters which all have the same color up; (2) Reverse several counters which all have the same color up. The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.

Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$ .

A point $P$ is given inside a triangle $ABC$ . Let $D,E,F$ be the midpoints of $AP,BP,CP$ , and let $L,M,N$ be the intersection points of $BF$ and $CE, AF$ and $CD, AE$ and $BD$ , respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$ . (b) Prove that $DL,EM$ , and $FN$ are concurrent.

An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most $\sqrt2$ , or there is a point on the distance $1/2$ from the edge of the board.

In a quadrilateral $ABCD$ with $AB // CD$ , the external bisectors of the angles at $B$ and $C$ meet at $P$ , while the external bisectors of the angles at $A$ and $D$ meet at $Q$ . Prove that the length of $PQ$ equals the semiperimeter of $ABCD$ .