1991 IMO Problems
Problems of the 1991 IMO.
Contents
Day I
Problem 1
Given a triangle let be the center of its inscribed circle. The internal bisectors of the angles meet the opposite sides in respectively. Prove that
Problem 2
Let be an integer and be all the natural numbers less than and relatively prime to . If prove that must be either a prime number or a power of .
Problem 3
Let . Find the smallest integer such that each -element subset of contains five numbers which are pairwise relatively prime.
Day II
Problem 4
Suppose is a connected graph with edges. Prove that it is possible to label the edges in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
Problem 5
Let be a triangle and an interior point of . Show that at least one of the angles is less than or equal to .
Problem 6
An infinite sequence of real numbers is said to be bounded if there is a constant such that for every . Given any real number construct a bounded infinite sequence such that for every pair of distinct nonnegative integers .
- 1991 IMO
- IMO 1991 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1991 IMO (Problems) • Resources | ||
Preceded by 1990 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1992 IMO |
All IMO Problems and Solutions |