IMO 2004 (XLV International Mathematical Olympiad)

Circumcircles of BMR and CNR meet on BC

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 $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

Topic: Geometria piana Metodo: Simmetria Abilita: Ragionamento geometrico, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Circumcircles of BMR and CNR meet on BC

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 $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

src_imho_2004__Q01

Polynomial with real roots satisfying functional relations

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

Topic: Algebra, Equazioni funzionali Metodo: Fattorizzazione, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern Area: Algebra e Analisi Fonte: apri PDF p.1

Polynomial with real roots satisfying functional relations

Find all polynomials $f$ with real coefficients such that for all real $a,b,c$ such that $ab+bc+ca=0$ we have the following relationships: $f(a-b)+f(b-c)+f(c-a)=2f(a+b+c).$

src_imho_2004__Q02

Covering m×n rectangles with hook-shaped hexominoes

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 \begin{itemize} \item the rectangle is covered without gaps and without overlaps \item no part of a hook covers area outside the rectangle. \end{itemize}

Topic: Combinatoria Metodo: Colorazione, Invarianti, Casework Abilita: Ragionamento geometrico, Conteggio sistematico, Modellizzazione Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Covering m×n rectangles with hook-shaped hexominoes

Defines 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 \begin{itemize} \item the rectangle is covered without gaps and without overlaps \item no part of a hook covers area outside the rectangle. I’m going to tell you.

src_imho_2004__Q03

Positive reals as triangle sides from sum-reciprocal inequality

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

Topic: Disuguaglianze, Teoria dei Numeri Metodo: Disuguaglianze, Casework, Estremalità Abilita: Manipolazione algebrica, Stima, Lettura attenta Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Positive reals as triangle sides from sum-reciprocal inequality

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

src_imho_2004__Q04

Cyclic quadrilateral iff AP=CP via angle bisectors

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

Topic: Geometria piana Metodo: Simmetria, Trigonometria Abilita: Ragionamento geometrico, Lettura attenta, Manipolazione algebrica Area: Geometria Fonte: apri PDF p.1

Cyclic quadrilateral if AP=CP by angle bisectors

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

src_imho_2004__Q05

Find all n with an alternating multiple

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.

Topic: Teoria dei Numeri, Combinatoria Metodo: Congruenze, Casework, method_casework Abilita: Riconoscimento di pattern, Conteggio sistematico, Modellizzazione Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Find all n with an alternating multiple

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.

src_imho_2004__Q06