1994 IMO Problems
Problems of the 1994 IMO.
Contents
Day I
Problem 1
Let and be two positive integers. Let , , , be different numbers from the set such that for any two indices and with and , there exists an index such that . Show that .
Problem 2
Let be an isosceles triangle with . is the midpoint of and is the point on the line such that is perpendicular to . is an arbitrary point on different from and . lies on the line and lies on the line such that are distinct and collinear. Prove that is perpendicular to if and only if .
Problem 3
For any positive integer , let be the number of elements in the set whose base 2 representation has precisely three s.
- (a) Prove that, for each positive integer , there exists at least one positive integer such that .
- (b) Determine all positive integers for which there exists exactly one with .
Day II
Problem 4
Find all ordered pairs where and are positive integers such that is an integer.
Problem 5
Let be the set of real numbers strictly greater than . Find all functions satisfying the two conditions:
1. for all and in ;
2. is strictly increasing on each of the intervals and .
Problem 6
Show that there exists a set of positive integers with the following property: For any infinite set of primes there exist two positive integers and each of which is a product of distinct elements of for some .
- 1994 IMO
- IMO 1994 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1994 IMO (Problems) • Resources | ||
Preceded by 1993 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1995 IMO |
All IMO Problems and Solutions |