1996 IMO Problems
Problems of the 1996 IMO.
Contents
Day I
Problem 1
We are given a positive integer and a rectangular board with dimensions , . The rectangle is divided into a grid of unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is . The task is to find a sequence of moves leading from the square with as a vertex to the square with as a vertex.
(a) Show that the task cannot be done if is divisible by or .
(b) Prove that the task is possible when .
(c) Can the task be done when ?
Problem 2
Let be a point inside triangle such that
Let , be the incenters of triangles , , respectively. Show that , , meet at a point.
Problem 3
Let denote the set of nonnegative integers. Find all functions from to itself such that
Day II
Problem 4
The positive integers and are such that the numbers and are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Problem 5
Let be a convex hexagon such that is parallel to , is parallel to , and is parallel to . Let , , denote the circumradii of triangles , , , respectively, and let denote the perimeter of the hexagon. Prove that
Problem 6
Let be three positive integers with . Let be an -tuple of integers satisfying the following conditions:
(i) ;
(ii) For each with , either or .
Show that there exists indices with , such that .
See Also
1996 IMO (Problems) • Resources | ||
Preceded by 1995 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1997 IMO |
All IMO Problems and Solutions |