2013 IMO Problems
Problem 1
Prove that for any pair of positive integers and
, there exist
positive integers
(not necessarily different) such that
Problem 2
A configuration of points in the plane is called Colombian if it consists of
red points and
blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian
configuration if the following two conditions are satisfied:
- no line passes through any point of the configuration;
- no region contains points of both colours.
Find the least value of such that for any Colombian configuration of
points, there is a good
arrangement of
lines.
Problem 3
Let the excircle of triangle opposite the vertex
be tangent to the side
at the point
. Define the points
on
and
on
analogously, using the excircles opposite
and
, respectively. Suppose that the circumcentre of triangle
lies on the circumcircle of triangle
. Prove that triangle
is right-angled.
Problem 4
Let be an acute triangle with orthocenter
, and let
be a point on the side
, lying strictly between
and
. The points
and
are the feet of the altitudes from
and
, respectively. Denote by
is [sic] the circumcircle of
, and let
be the point on
such that
is a diameter of
. Analogously, denote by
the circumcircle of triangle
, and let
be the point such that
is a diameter of
. Prove that
and
are collinear.
Problem 5
Let be the set of all positive rational numbers. Let
be a function satisfying the following three conditions:
(i) for all , we have
;
(ii) for all
, we have
;
(iii) there exists a rational number
such that
.
Prove that for all
.
Problem 6
Let be an integer, and consider a circle with
equally spaced points marked on it. Consider all labellings of these points with the numbers
such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels
with
, the chord joining the points labelled
and
does not intersect the chord joining the points labelled
and
.
Let be the number of beautiful labelings, and let N be the number of ordered pairs
of positive integers such that
and
. Prove that
2013 IMO (Problems) • Resources | ||
Preceded by 2012 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2014 IMO Problems |
All IMO Problems and Solutions |