1997 IMO Problems
Problems of the 1997 IMO.
Contents
Day I
Problem 1
In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively black and white (as on a chessboard).
For any pair of positive integers and , consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths and , lie along edges of the squares.
Let be the total area of the black part of the triangle and be the total area of the white part.
Let
(a) Calculate for all positive integers and which are either both even or both odd.
(b) Prove that for all and .
(c) Show that there is no constant such that for all and .
Problem 2
The angle at is the smallest angle of triangle . The points and divide the circumcircle of the triangle into two arcs. Let be an interior point of the arc between and which does not contain . The perpendicular bisectors of and meet the line and and , respectively. The lines and meet at . Show that.
Problem 3
Let , ,..., be real numbers satisfying the conditions
and
, for
Show that there exists a permutation , ,..., of , ,..., such that
Day II
Problem 4
An matrix whose entries come from the set is called a matrix if, for each , the th row and the th column together contain all elements of . Show that
(a) there is no matrix for ;
(b) matrices exist for infinitely many values of .
Problem 5
Find all pairs of integers that satisfy the equation
Problem 6
For each positive integer , let denote the number of ways of representing as a sum of powers of with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, , because the number 4 can be represented in the following four ways:
Prove that, for any integer ,
.
See Also
1997 IMO (Problems) • Resources | ||
Preceded by 1996 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1998 IMO |
All IMO Problems and Solutions |