1991 IMO Problems
Problems of the 1991 IMO.
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
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 .
Let . Find the smallest integer such that each -element subset of contains five numbers which are pairwise relatively prime.
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.
Let be a triangle and an interior point of . Show that at least one of the angles is less than or equal to .
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
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