1963 IMO Problems
Problems of the 5th IMO 1963 Poland.
Contents
Day I
Problem 1
Find all real roots of the equation
where is a real parameter.
Problem 2
Point and segment are given. Determine the locus of points in space which are the vertices of right angles with one side passing through , and the other side intersecting the segment .
Problem 3
In an -gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
Prove that .
Day II
Problem 4
Find all solutions of the system where is a parameter.
Problem 5
Prove that .
Problem 6
Five students, , took part in a contest. One prediction was that the contestants would finish in the order . This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order . This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
Resources
- 1963 IMO
- IMO 1963 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1963 IMO (Problems) • Resources | ||
Preceded by 1962 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1964 IMO |
All IMO Problems and Solutions |