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:
![\[k < \frac {1}{2} + \sqrt {2 \cdot n}\]](http://latex.artofproblemsolving.com/e/2/5/e2599512618364e8e8a4e77ea1fec25d1fa561fb.png)
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:
![\[\frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} + \frac {1}{\sqrt {BC}}\]](http://latex.artofproblemsolving.com/c/e/8/ce8f28be318ef82e52c506a605fb69133761ee67.png)
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
| 1989 IMO (Problems) • Resources | ||
| Preceded by 1988 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1990 IMO |
| All IMO Problems and Solutions | ||
