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

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/AB = t, find EG/EF in terms of t.

2. Let n ³ 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.

July 13, 1990
Day II

4. Let Q⁺ be the set of positive rational numbers. Construct a function f : Q⁺ ® Q⁺ such that f(xf(y)) =
f(x)/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 n_2k, A chooses any integer n_2k+1 such that n_2k £ n_2k+1 £ (n_2k)².
Knowing n_2k+1, B chooses any integer n_2k+2 such that (n_2k+1)/(n_2k+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.