17th Mexican 2003


A1. Find all positive integers with two or more digits such that if we insert a 0 between the units and tens digits we get a multiple of the original number.
A2. A, B, C are collinear with B betweeen A and C. K₁ is the circle with diameter AB, and K₂ is the circle with diameter BC. Another circle touches AC at B and meets K₁ again at P and K₂ again at Q. The line PQ meets K₁ again at R and K₂ again at S. Show that the lines AR and CS meet on the perpendicular to AC at B.
A3. At a party there are n women and n men. Each woman likes r of the men, and each man likes r of then women. For which r and s must there be a man and a woman who like each other?
B1. The quadrilateral ABCD has AB parallel to CD. P is on the side AB and Q on the side CD such that AP/PB = DQ/CQ. M is the intersection of AQ and DP, and N is the intersection of PC and QB. Find MN in terms of AB and CD.
B2. Some cards each have a pair of numbers written on them. There is just one card for each pair (a,b) with 1 ≤ a < b ≤ 2003. Two players play the following game. Each removes a card in turn and writes the product ab of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to 1 loses. Which player has a winning strategy?
B3. Given a positive integer n, an allowed move is to form 2n+1 or 3n+2. The set S_n is the set of all numbers that can be obtained by a sequence of allowed moves starting with n. For example, we can form 5 → 11 → 35 so 5, 11 and 35 belong to S₅. We call m and n compatible if S_m ∩ S_n is non-empty. Which members of {1, 2, 3, ... , 2002} are compatible with 2003?

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