2002 IMO Problems
Problems of the 2002 IMO.
Contents
Day I
Problem 1
is the set of all
with
non-negative integers such that
. Each element of
is colored red or blue, so that if
is red and
, then
is also red. A type
subset of
has
blue elements with different first member and a type
subset of
has
blue elements with different second member. Show that there are the same number of type
and type
subsets.
Problem 2
is a diameter of a circle center
.
is any point on the circle with
.
is the chord which is the perpendicular bisector of
.
is the midpoint of the minor arc
. The line through
parallel to
meets
at
. Show that
is the incenter of triangle
.
Problem 3
Find all pairs of positive integers for which here exist infinitely many positive integers
such that
is itself an integer.
Day II
Problem 4
Let be an integer and let
be all of its positive divisors in increasing order. Show that
Problem 5
Find all functions such that
for all real numbers .
Problem 6
Let be a positive integer. Let
be unit circles in the plane, with centers
respectively. If no line meets more than two of the circles, prove that
See Also
2002 IMO (Problems) • Resources | ||
Preceded by 2001 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2003 IMO |
All IMO Problems and Solutions |