13th Mexican 1999


A1. 1999 cards are lying on a table. Each card has a red side and a black side and can be either side up. Two players play alternately. Each player can remove any number of cards showing the same color from the table or turn over any number of cards of the same color. The winner is the player who removes the last card. Does the first or second player have a winning strategy?
A2. Show that there is no arithmetic progression of 1999 distinct positive primes all less than 12345.
A3. P is a point inside the triangle ABC. D, E, F are the midpoints of AP, BP, CP. The lines BF, CE meet at L; the lines CD, AF meet at M; and the lines AE, BD meet at N. Show that area DNELFM = (1/3) area ABC. Show that DL, EM, FN are concurrent.
B1. 10 squares of a chessboard are chosen arbitrarily and the center of each chosen square is marked. The side of a square of the board is 1. Show that either two of the marked points are a distance ≤ √2 apart or that one of the marked points is a distance 1/2 from the edge of the board.
B2. ABCD has AB parallel to CD. The exterior bisectors of ∠B and ∠C meet at P, and the exterior bisectors of ∠A and ∠D meet at Q. Show that PQ is half the perimeter of ABCD.
B3. A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping 2 x 1 dominos, then at least one of its sides has even length.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.