国际数学奥林匹克IMO试题(官方版)1990_eng.pdf

31stInternational 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 AB = t, fi nd 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. 31stInternational 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(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 n2k, A chooses any integer n2k+1such that n2k n2k+1 n2 2k. Knowing n2k+1, B chooses any integer n2k+2such 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 n0does: (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, ., 19902in some order. 。