45^{th} INTERNATIONAL MATHEMATICAL OLYMPIAD IMO 2004 HELLAS 



Day: 
1 
Country Code: 
66 
Country Abbr.: 
SIN 
Language: 
English 



Problem 1. Let be an acuteangled 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 which satisfy the equality
for all real numbers such that .
Problem 3. Define a hook to be a figure made up of six unit squares as shown in the diagram









or any of the figures obtained by applying rotations and reflections to this figure.
Determine all rectangles that can be covered with hooks so that
· the rectangle is covered without gaps and without overlaps;
· no part of a hook covers area outside the rectangle.
Duration of exam: 4 hours 30 minutes.
Each problem is worth 7 marks.