International Mathematical Olympiad

31st International Mathematical Olympiad Beijing, China Day I July 12, 1990

1. Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent line at E to the circle through D, E, and M intersects the lines BC and AC at F and G, respectively. If AM = t, AB find EG EF in terms of t. 2. Let n ≥ 3 and consider a set E of 2n − 1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E. Find the smallest value of k so that every such coloring of k points of E is good. 3. Determine all integers n > 1 such that 2n + 1 n2 is an integer.

31st International Mathematical Olympiad Beijing, China Day II July 13, 1990

4. Let Q+ be the set of positive rational numbers. Construct a function f : Q+ → Q+ such that f (x) f (xf (y)) = y for all x, y in Q+ . 5. Given an initial integer n0 > 1, two players, A and B, choose integers n1 , n2 , n3 , . . . alternately according to the following rules: Knowing n2k , A chooses any integer n2k+1 such that n2k ≤ n2k+1 ≤ n2 . 2k Knowing n2k+1 , B chooses any integer n2k+2 such that n2k+1 n2k+2 is a prime raised to a positive integer power. Player A wins the game by choosing the number 1990; player B wins by choosing the number 1. For which n0 does: (a) A have a winning strategy? (b) B have a winning strategy? (c) Neither player have a winning strategy? 6. Prove that there exists a convex 1990-gon with the following two properties: (a) All angles are equal. (b) The lengths of the 1990 sides are the numbers 12 , 22 , 32 , . . . , 19902 in some order.