1984 IMO Problems
Problems of the 1984 IMO.
Prove that , where and are non-negative real numbers satisfying .
Find one pair of positive integers such that is not divisible by , but is divisible by .
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 .
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.
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 .
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
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