2007 IMO Problems
Problem 1
Real numbers are given. For each
(
) define
and let
(a) Prove that, for any real numbers ,
(b) Show that there are real numbers such that equality holds in (*)
Problem 2
Consider five points , and
such that
is a parallelogram and
is a cyclic quadrilateral.
Let
be a line passing through
. Suppose that
intersects the interior of the segment
at
and intersects
line
at
. Suppose also that
. Prove that
is the bisector of
.
Problem 3
In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.
Problem 4
In the bisector of
intersects the circumcircle again at
, the perpendicular bisector of
at
, and the perpendicular bisector of
at
. The midpoint of
is
and the midpoint of
is
. Prove that the triangles
and
have the same area.
Problem 5
(Kevin Buzzard and Edward Crane, United Kingdom)
Let and
be positive integers. Show that if
divides
, then
.
Problem 6
Let be a positive integer. Consider
as a set of
points in three-dimensional space.
Determine the smallest possible number of planes, the union of which contain
but does not include
.
2007 IMO (Problems) • Resources | ||
Preceded by 2006 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2008 IMO Problems |
All IMO Problems and Solutions |