International Competitions 2004

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 $BC$.

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 $$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$

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.

Let $n \geq 3$ be an integer. Let $t_1, t_2, \ldots, t_n$ be positive real numbers such that $$n^2 + 1 > (t_1 + t_2 + \ldots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \ldots + \frac{1}{t_n}\right).$$

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

In a convex quadrilateral $ABCD$ the diagonal $BD$ does not bisect the angles $\angle ABC$ and $\angle CDA$. The 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 $AP = CP$.

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.