30^th International Mathematical Olympiad
Braunschweig, Germany
Day I

1. Prove that the set {1, 2, . . . , 1989} can be expressed as the disjoint union of subsets A_i (i = 1, 2, . . . , 117) such that:
(i) Each A_i contains 17 elements;
(ii) The sum of all the elements in each A_i is the same.

2. In an acute-angled triangle ABC the internal bisector of angle A meets the circumcircle of the triangle again at A₁. Points B₁ and C₁ are defined similarly. Let A₀ be the point of intersection of the line AA₁ with the external bisectors of angles B and C. Points B₀ and C₀ are defined similarly. Prove that:
(i) The area of the triangle A₀B₀C₀ is twice the area of the hexagon AC₁BA₁CB₁.
(ii) The area of the triangle A₀B₀C₀ is at least four times the area of the triangle ABC.

3. Let n and k be positive integers and let S be a set of n points in the plane such that
(i) No three points of S are collinear, and
(ii) For any point P of S there are at least k points of S equidistant from P.
Prove that: k < 1/2 + Ö2n

Day II

4. Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that (1/Ö h) ³ (1/ÖAD + 1/ÖBC).

5. Prove that for each positive integer n there exist n consecutive positive integers none of which is an integral power of a prime number.

6. A permutation (x₁, x₂, . . . , x_m) of the set {1, 2, . . . , 2n}, where n is a positive integer, is said to have property P if |x_i−x_i+1|=n for at least one i in {1, 2, . . . , 2n − 1}. Show that, for each n, there are more permutations with property P than without.