国际数学奥林匹克IMO试题(官方版)1987_eng.pdf

28thInternational Mathematical Olympiad Havana, Cuba Day I July 10, 1987 1.Let pn(k) be the number of permutations of the set 1,.,n, n 1, which have exactly k fi xed points. Prove that n X k=0 k pn(k) = n!. (Remark: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S is called a fi xed point of the permutation f if f(i) = i.) 2.In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas. 3.Let x1, x2, ., xnbe real numbers satisfying x2 1+x 2 2+x 2 n = 1. Prove that for every integer k 2 there are integers a1, a2, ., an, not all 0, such that |ai| k 1 for all i and |a1x1+ a1x2+ + anxn| (k 1)n kn 1 . 28thInternational Mathematical Olympiad Havana, Cuba Day II July 11, 1987 4.Prove that there is no function f from the set of non-negative integers into itself such that f(f(n) = n + 1987 for every n. 5.Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. 6.Let n be an integer greater than or equal to 2. Prove that if k2+k+n is prime for all integers k such that 0 k pn/3, then k2 + k + n is prime for all integers k such that 0 k n 2. 。