13th Iberoamerican Mathematical Olympiad Problems 1998

A1.  There are 98 points on a circle. Two players play alternately as follows. Each player joins two points which are not already joined. The game ends when every point has been joined to at least one other. The winner is the last player to play. Does the first or second player have a winning strategy?

A2.  The incircle of the triangle ABC touches BC, CA, AB at D, E, F respectively. AD meets the circle again at Q. Show that the line EQ passes through the midpoint of AF iff AC = BC.

A3.  Find the smallest number n such that given any n distinct numbers from {1, 2, 3, ... , 999}, one can choose four different numbers a, b, c, d such that a + 2b + 3c = d.

B1.  Representatives from n > 1 different countries sit around a table. If two people are from the same country then their respective right hand neighbors are from different countries. Find the maximum number of people who can sit at the table for each n.

B2.  P₁, P₂, ... , P_n are points in the plane and r₁, r₂, ... , r_n are real numbers such that the distance between P_i and P_j is r_i + r_j (for i not equal to j). Find the largest n for which this is possible.

B3.  k is the positive root of the equation x² - 1998x - 1 = 0. Define the sequence x₀, x₁, x₂, ... by x₀ = 1, x_n+1 = [k x_n]. Find the remainder when x₁₉₉₈ is divided by 1998.