2004 IMO Problems

Problems of the 45th IMO 2004 Athens, Greece.

Day 1

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 $BC$.

Solution

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

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

Solution

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.

[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; (a) the rectangle is covered without gaps and without overlaps, (b) no part of a hook covers area outside the rectangle.

Solution

Day 2

Problem 4

Let $n \geq 3$ be an integer. Let $t_1, t_2, \dots ,t_n$ be positive real numbers such that

\[n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

Solution

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

Solution

Problem 6

We call a positive integer alternating if every two consecutive digits in its decimal representation have a different parity.

Find all positive integers $n$ such that $n$ has a multiple which is alternating.

Solution

Resources

2004 IMO (Problems) • Resources
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2003 IMO Problems
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2005 IMO Problems
All IMO Problems and Solutions
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