IMO 2009 (L International Mathematical Olympiad)

Divisibility condition on sequence elements

Let $n$ be a positive integer and let $a_1,a_2,\dots ,a_k$ ($k\ge 2$) be distinct integers in the set $\{1,2,\dots ,n\}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,2,\dots ,k-1$. Prove that $n$ does not divide $a_k(a_1-1)$.

Topic: Teoria dei Numeri, Combinatoria Metodo: Invarianti, Congruenze Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Divisibility condition on sequence elements

Let $n$ be a positive integer and let $a_1,a_2,\dots ,a_k$ ($k\ge 2$) be distinct integers in the set $\{1,2,\dots ,n\}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,2,\dots ,k-1$. Prove that $n$ does not divide $a_k(a_1-1)$.

src_imho_2009__Q01

Circle tangent to PQ in triangle with circumcircle

Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. Let $K$, $L$, and $M$ be the midpoints of the segments $BP$, $CQ$, and $PQ$, respectively, and let $\Gamma$ be the circle passing through $K$, $L$, and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP=OQ$.

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

Circle tangent to PQ in triangle with circumcircle

Let $ABC$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. Let $K$, $L$, and $M$ be the midpoints of the segments $BP$, $CQ$, and $PQ$, respectively, and let $\Gamma$ be the circle passing through $K$, $L$, and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that$OP=OQ$.

src_imho_2009__Q02

Arithmetic progressions in subsequences imply full AP

Suppose that $s_1,s_2,s_3,\dots$ is a strictly increasing sequence of positive integers such that the subsequences $s_{s_1},s_{s_2},s_{s_3},\dots \text{and}{s}_{s_1+1},s_{s_2+1},s_{s_3+1},\dots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\dots$ is an arithmetic progression.

Topic: Algebra, Combinatoria Metodo: Induzione, Ricorsione, Telescoping Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Arithmetic progressions in subsequences imply full AP

Suppose that $s_1,s_2,s_3,\dots$ is a strictly increasing sequence of positive integers such that the subsequences $s_{s_1},s_{s_2},s_{s_3},\dots \text{and}{s}_{s_1+1},s_{s_2+1},s_{s_3+1},\dots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\dots$ is an arithmetic progression.

src_imho_2009__Q03

Find all angles CAB in triangle with angle bisectors and incenter

Let $ABC$ be a triangle with $AB=AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle BEK=45^{\circ }$. Find all possible values of $\angle CAB$.

Topic: Geometria piana, Trigonometria Metodo: Trigonometria, Casework Abilita: Ragionamento geometrico, Manipolazione algebrica, Casework accurato Area: Geometria Fonte: apri PDF p.1

Find all angles CAB in triangle with angle bisectors and incenter

Let $ABC$ be a triangle with $AB=AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK=45^{\circ }$. Find all possible values of $\angle CAB$.

src_imho_2009__Q04

Functional equation over positive integers with triangle condition

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $a,f(b)\text{and}f(b+f(a)-1).$ (A triangle is non-degenerate if its vertices are not collinear.)

Topic: Equazioni funzionali, Algebra Metodo: Casework, Estremalità, Induzione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Casework accurato, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Functional equation over positive integers with triangle condition

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $a,f(b)\text{and}f(b+f(a)-1).$ (A triangle is non-degenerate if its vertices are not collinear.)

src_imho_2009__Q05

Grasshopper jumps avoid finite set on positive real axis

Let $a_1,a_2,\dots ,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\cdots +a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1,a_2,\dots ,a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

Topic: Combinatoria Metodo: Invarianti, Estremalità, Induzione, Grafi Abilita: Modellizzazione, Ragionamento geometrico, Conteggio sistematico, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Grasshopper jumps avoid finite set on positive real axis

Let $a_1,a_2,\dots ,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\cdots +a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1,a_2,\dots ,a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

src_imho_2009__Q06