IMO 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 $BC$ .

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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

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

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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.

[asy] unitsize(0.5 cm);

draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] 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 $t_1$ , $t_2$ , ..., $t_n$ 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 $1 \leq i < j < k \leq n$ .

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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 $$\angle PBC=\angle DBA \text{and} \angle PDC=\angle BDA.$$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$ .

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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.

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