1989 IMO Problems
Problems of the 1989 IMO.
Contents
Day I
Problem 1
Prove that in the set can be expressed as the disjoint union of subsets such that
i.) each contains 17 elements
ii.) the sum of all the elements in each is the same.
Problem 2
is a triangle, the bisector of angle meets the circumcircle of triangle in , points and are defined similarly. Let meet the lines that bisect the two external angles at and in . Define and similarly. Prove that the area of triangle area of hexagon area of triangle .
Problem 3
Let and be positive integers and let be a set of points in the plane such that
i.) no three points of are collinear, and
ii.) for every point of there are at least points of equidistant from
Prove that:
Day II
Problem 4
Let be a convex quadrilateral such that the sides satisfy There exists a point inside the quadrilateral at a distance from the line such that and Show that:
Problem 5
Prove that for each positive integer there exist consecutive positive integers none of which is an integral power of a prime number.
Problem 6
A permutation of the set where is a positive integer, is said to have property if for at least one in Show that, for each , there are more permutations with property than without.
- 1989 IMO
- IMO 1989 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1979 IMO (Problems) • Resources | ||
Preceded by 1988 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1990 IMO |
All IMO Problems and Solutions |