1997 IMO Problems
Problems of the 1997 IMO.
Contents
[hide]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 |