1999 IMO Problems
Problems of the 1999 IMO.
Contents
[hide]Day I
Problem 1
Determine all finite sets of at least three points in the plane which satisfy the following condition:
For any two distinct points and in , the perpendicular bisector of the line segment is an axis of symmetry of .
Problem 2
Let be a fixed integer.
- (a) Find the least constant such that for all nonnegative real numbers ,
- (b) Determine when equality occurs for this value of .
Problem 3
Consider an square board, where is a fixed even positive integer. The board is divided into units squares. We say that two different squares on the board are adjacent if they have a common side.
unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.
Determine the smallest possible value of .
Day II
Problem 4
Determine all pairs of positive integers such that
is a prime, not exceeded , and is divisible by .
Problem 5
Two circles and are contained inside the circle , and are tangent to at the distinct points and , respectively. passes through the center of . The line passing through the two points of intersection of and meets at and . The lines and meet at and , respectively.
Prove that is tangent to .
Problem 6
Determine all functions such that
for all real numbers .
See Also
1999 IMO (Problems) • Resources | ||
Preceded by 1998 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2000 IMO |
All IMO Problems and Solutions |