IMO 2004

Problem 1. Let $ABC$ be an acute-angled triangle with $AB\ne AC$. 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 $BC$.

Problem 2. Find all polynomials $P(x)$ with real coefficients which satisfy the equality

$P(a-b)+P(b-c)+P(c-a)=2P(a+b+c)$

for all real numbers $a$, $b$, $c$ such that $ab+bc+ca=0$.

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 $m\times n$ 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.

Problem 4. Let $n\ge 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that

Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1\le i<j<k\le n$.

Problem 5. 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

$\angle PBC=\angle DBA\text{and}\angle PDC=\angle BDA.$

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.