5th OMCC 2003


A1. There are 2003 stones in a pile. Two players alternately select a positive divisor of the number of stones currently in the pile and remove that number of stones. The player who removes the last stone loses. Find a winning strategy for one of the players.
A2. AB is a diameter of a circle. C and D are points on the tangent at B on opposite sides of B. AC, AD meet the circle again at E, F respectively. CF, DE meet the circle again at G, H respectively. Show that AG = AH.
A3. Given integers a > 1, b > 2, show that a^b + 1 ≥ b(a+1). When do we have equality?
B1. Two circles meet at P and Q. A line through P meets the circles again at A and A'. A parallel line through Q meets the circles again at B and B'. Show that PBB' and QAA' have equal perimeters.
B2. An 8 x 8 board is divided into unit squares. Each unit square is painted red or blue. Find the number of ways of doing this so that each 2 x 2 square (of four unit squares) has two red squares and two blue squares.
B3. Call a positive integer a tico if the sum of its digits (in base 10) is a multiple of 2003. Show that there is an integer N such that N, 2N, 3N, ... , 2003N are all ticos. Does there exist a positive integer such that all its multiples are ticos?

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