1984 IMO Problems
Problems of the 1984 IMO.
Contents
Day I
Problem 1
Prove that , where and are non-negative real numbers satisfying .
Problem 2
Find one pair of positive integers such that is not divisible by , but is divisible by .
Problem 3
Given points and in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point in the plane, the circle has center and radius , where is measured in radians in the range . Prove that we can find a point , not on , such that its color appears on the circumference of the circle .
Day II
Problem 4
Let be a convex quadrilateral with the line being tangent to the circle on diameter . Prove that the line is tangent to the circle on diameter if and only if the lines and are parallel.
Problem 5
Let be the sum of the lengths of all the diagonals of a plane convex polygon with vertices (where ). Let be its perimeter. Prove that: where denotes the greatest integer not exceeding .
Problem 6
Let be odd integers such that and . Prove that if and for some integers and , then .
- 1984 IMO
- IMO 1984 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1979 IMO (Problems) • Resources | ||
Preceded by 1983 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1985 IMO |
All IMO Problems and Solutions |