28th International Mathematical Olympiad
Havana, Cuba
July 10, 1987
Day I
 

1. Let p_n(k) be the number of permutations of the set {1, . . . , n}, n ³ 1, which have exactly k fixed
points. Prove that (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 fixed point of the permutation f if f(i) = i.)

2. In an acute-angled triangle ABC the interior bisector of the angle A intersectsBC 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, . . . , x_n be real numbers satisfying x₁²+x₂²+· · ·+x_n² = 1. Prove that for every integer k ³ 2 there are integers a₁, a₂, . . . , a_n, not all 0, such that |a_i|£ k - 1 for all i and

July 11, 1987
Day II
 

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 £ Ön/3, then k² + k + n is prime for all integers k such that 0 £ k £ n - 2.