International Mathematical Olympiad  – IMO 2004 Problems

Problem 1

Let $ABC$ be an acute-angled triangle with $AB\neq AC$. 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 $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

Problem 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

\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

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 $ m\times 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.

Problem 4

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, …, $t_n$ be positive real numbers such that\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

Problem 5

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies\[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\]Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

Problem 6

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.