14th Mexican 2000


A1. A, B, C, D are circles such that A and B touch externally at P, B and C touch externally at Q, C and D touch externally at R, and D and A touch externally at S. A does not intersect C, and B does not intersect D. Show that PQRS is cyclic. If A and C have radius 2, B and D have radius 3, and the distance between the centers of A and C is 6, find area PQRS.
A2. A triangle is constructed like that below, but with 1, 2, 3, ... , 2000 as the first row. Each number is the sum of the two numbers immediately above. Find the number at the bottom of the triangle. 1 2 3 4 5 3 5 7 9 8 12 16 20 28 48
A3. If A is a set of positive integers, take the set A' to be all elements which can be written as ± a₁ ± a₂ ... ± a_n, where a_i are distinct elements of A. Similarly, form A" from A'. What is the smallest set A such that A" contains all of 1, 2, 3, ... , 40?
B1. Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows: a₁ = 5, a_n+1 = a a_n + b. What is the largest number of primes that can be obtained before the first composite member of the sequence?
B2. Given an n x n board with squares colored alternately black and white like a chessboard. An allowed move is to take a rectangle of squares (with one side greater than one square, and both sides odd or both sides even) and change the color of each square in the rectangle. For which n is it possible to end up with all the squares the same color by a sequence of allowed moves?
B3. ABC is a triangle with ∠B > 90^o. H is a point on the side AC such that AH = BH and BH is perpendicular to BC. D, E are the midpoints of AB, BC. The line through H parallel to AB meets DE at F. Show that ∠BCF = ∠ACD.

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