16th Iberoamerican 2001

16th Iberoamerican 2001 problems


A1. Show that there are arbitrarily large numbers n such that: (1) all its digits are 2 or more; and (2) the product of any four of its digits divides n.
A2. ABC is a triangle. The incircle has center I and touches the sides BC, CA, AB at D, E, F respectively. The rays BI and CI meet the line EF at P and Q respectively. Show that if DPQ is isosceles, then ABC is isosceles.
A3. Let X be a set with n elements. Given k > 2 subsets of X, each with at least r elements, show that we can always find two of them whose intersection has at least r - nk/(4k - 4) elements.
B1. Call a set of 3 distinct elements which are in arithmetic progression a trio. What is the largest number of trios that can be subsets of a set of n distinct real numbers?
B2. Two players play a game on a 2000 x 2001 board. Each has one piece and the players move their pieces alternately. A short move is one square in any direction (including diagonally) or no move at all. On his first turn each player makes a short move. On subsequent turns a player must make the same move as on his previous turn followed by a short move. This is treated as a single move. The board is assumed to wrap in both directions so a player on the edge of the board can move to the opposite edge. The first player wins if he can move his piece onto the same square as his opponent's piece. For example, suppose we label the squares from (0, 0) to (1999, 2000), and the first player's piece is initially at (0, 0) and the second player's at (1996, 3). The first player could move to (1999, 2000), then the second player to (1996, 2). Then the first player could move to (1998, 1998), then the second player to (1995, 1). Can the first player always win irrespective of the initial positions of the two pieces?
B3. Show that a square with side 1 cannot be covered by five squares with side less than 1/2.

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