45th INTERNATIONAL MATHEMATICAL OLYMPIAD

IMO 2004 HELLAS

 

Day:

 1

Country Code:

 66

Country Abbr.:

SIN

Language:

 English

 

                 

 

 

 

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 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.