2004 IMO Problems
Problems of the 45th IMO 2004 Athens, Greece.
Contents
Day 1
Problem 1
Let be an acute-angled triangle with . The circle with diameter intersects the sides and at and respectively. Denote by the midpoint of the side . The bisectors of the angles and intersect at . Prove that the circumcircles of the triangles and have a common point lying on the side .
Problem 2
Find all polynomials with real coefficients such that for all reals such that we have the following relations
Problem 3
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
Determine all rectangles that can be covered without gaps and without overlaps with hooks such that; (a) the rectangle is covered without gaps and without overlaps, (b) no part of a hook covers area outside the rectangle.
Day 2
Problem 4
Let be an integer. Let be positive real numbers such that
Show that , , are side lengths of a triangle for all , , with .
Problem 5
In a convex quadrilateral , the diagonal bisects neither the angle nor the angle . The point lies inside and satisfies Prove that is a cyclic quadrilateral if and only if
Problem 6
We call a positive integer alternating if every two consecutive digits in its decimal representation have a different parity.
Find all positive integers such that has a multiple which is alternating.
Resources
2004 IMO (Problems) • Resources | ||
Preceded by 2003 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2005 IMO Problems |
All IMO Problems and Solutions |