| 45th International Mathematical Olympiad July 12th, 2004 Athens, Greece Day I |
| 1. Let ABC be an acute-angled triangle with AB 6= AC. The circle with diameter BC intersects the sides AB and AC at M and N respectively. Denote by O the midpoint of the side BC. The bisectors of the angles \BAC and \MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the side BC. |
 |
| 2. Find all polynomials f with real coefficients such that for all reals a, b, c such that ab + bc + ca = 0 we have the following relations f(a ? b) + f(b ? c) + f(c ? a) = 2f(a + b + c). |
| 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 m ? n rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle. |
| Work time: 4.5 hours. |
| 45th International Mathematical Olympiad July 13th, 2004 Athens, Greece Day II |
| 2. In a convex quadrilateral ABCD the diagonal BD bisects neither the angle ABC nor the angle CDA. A point P lies inside ABCD and satisfies \PBC = \DBA and \PDC = \BDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP. |
| 3. We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers n such that n has a multiple which is alternating. |
| Work time: 4.5 hours. |
| 1. Let n >= 3 be an integer. Let t1, t2, ..., tn be positive real numbers such that (n^2 +1) > (t1 + t2 + ... + tn) (1/t1 + 1/t2 + ... +1/tn) Show that ti, tj , tk are side lengths of a triangle for all i, j, k with 1 <= i < j < k <= n. |
| You can find more problems at : |