1987 IMO Problems
Problems of the 1987 IMO Cuba.
Contents
Day I
Problem 1
Let be the number of permutations of the set , which have exactly fixed points. Prove that
.
(Remark: A permutation of a set is a one-to-one mapping of onto itself. An element in is called a fixed point of the permutation if .)
Problem 2
In an acute-angled triangle the interior bisector of the angle intersects at and intersects the circumcircle of again at . From point perpendiculars are drawn to and , the feet of these perpendiculars being and respectively. Prove that the quadrilateral and the triangle have equal areas.
Problem 3
Let be real numbers satisfying . Prove that for every integer there are integers , not all 0, such that for all and
.
Day 2
Problem 4
Prove that there is no function from the set of non-negative integers into itself such that for every .
Problem 5
Let be an integer greater than or equal to 3. Prove that there is a set of points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Problem 6
Let be an integer greater than or equal to 2. Prove that if is prime for all integers such that , then is prime for all integers such that .
Resources
- 1987 IMO
- IMO 1987 problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1987 IMO (Problems) • Resources | ||
Preceded by 1986 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1988 IMO |
All IMO Problems and Solutions |