1992 IMO Problems
Problems of the 1992 IMO.
Contents
Day I
Problem 1
Find all integers , , satisfying such that is a divisor of .
Problem 2
Let denote the set of all real numbers. Find all functions such that
Problem 3
Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of such that whenever exactly n edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Day II
Problem 4
In the plane let be a circle, a line tangent to the circle , and a point on . Find the locus of all points with the following property: there exists two points , on such that is the midpoint of and is the inscribed circle of triangle .
Problem 5
Let be a finite set of points in three-dimensional space. Let ,,, be the sets consisting of the orthogonal projections of the points of onto the -plane, -plane, -plane, respectively. Prove that
where denotes the number of elements in the finite set .
Problem 6
For each positive integer , is defined to be the greatest integer such that, for every positive integer , can be written as the sum of positive squares.
(a) Prove that for each .
(b) Find an integer such that .
(c) Prove that there are infinitely many integers such that .
1992 IMO (Problems) • Resources | ||
Preceded by 1991 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1993 IMO |
All IMO Problems and Solutions |