IMO 2005 (XLVI International Mathematical Olympiad)

Points on equilateral triangle sides form convex hexagon with concurrent diagonals

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$; $B_1$, $B_2$ on $CA$; and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1{A}_2{B}_1{B}_2{C}_1{C}_2$ with equal side lengths. Prove that the lines $A_1{B}_2$, $B_1{C}_2$ and $C_1{A}_2$ are concurrent.

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

Points on equilateral triangle sides form convex hexagon with concurrent diagonals

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$; $B_1$, $B_2$ on $CA$; and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1{A}_2{B}_1{B}_2{C}_1{C}_2$ with equal side lengths. Prove that the lines $A_1{B}_2$, $B_1{C}_2$ and $C_1{A}_2$ are concurrent.

src_imho_2005__Q01

Integer sequence with n-divisibility; every integer occurs exactly once

Let $a_1,a_2,\dots$ be a sequence of integers with infinitely many positive and infinitely many negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\dots ,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence.

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

Integer sequence with n-divisibility; every integer occurs exactly once

Let $a_1,a_2,\dots$ be a sequence of integers with infinitely many positive and infinitely many negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\dots ,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence.

src_imho_2005__Q02

Inequality for x,y,z >= 1 involving fractional expressions

Let $x,y,z$ be real numbers each greater than $1$. Prove that $\frac{x^5-x^2}{x^5+y^3+z^3}+\frac{y^5-y^2}{y^5+z^3+x^3}+\frac{z^5-z^2}{z^5+x^3+y^3}\ge 0.$

Topic: Disuguaglianze, Algebra Metodo: Disuguaglianze, Simmetria Abilita: Manipolazione algebrica, Lettura attenta, Stima Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality for x,y,z >= 1 involving fractional expressions

Let $x,y,z$ be real numbers each greater than $1$. Prove that $\frac{x^5-x^2}{x^5+y^3+z^3}+\frac{y^5-y^2}{y^5+z^3+x^3}+\frac{z^5-z^2}{z^5+x^3+y^3}\ge 0.$

src_imho_2005__Q03

Find all integers relatively prime to all terms of sequence $a_n=2^n+3^n+6^n-1$

Determine all positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n-1,n\ge 1.$

Topic: Teoria dei Numeri, Combinatoria Metodo: Congruenze, Fattorizzazione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Find all integers relatively prime to all terms of sequence $a_n=2^n+3^n+6^n-1$

Determine the positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n-1,n\ge 1.$

src_imho_2005__Q04

Circumcircles of triangles PQB, EF, AC meet at point other than P

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie on the sides $BC$ and $DA$, respectively, and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQB$, $QRE$, and $PRA$ have a common point other than $P$.

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

Circumcircles of triangles PQB, EF, AC meet at point other than P

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie on the sides $BC$ and $DA$, respectively, and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQB$, $QRE$, and $PRA$ have a common point other than $P$.

src_imho_2005__Q05

Math competition: at least 2 contestants solved exactly 5 problems each

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac{2}{5}$ of the contestants. Moreover, no contestant solved all the $6$ problems. Prove that there are at least $2$ contestants who solved exactly $5$ problems each.

Topic: Combinatoria Metodo: Doppio conteggio, Casework, Principio dei cassetti Abilita: Conteggio sistematico, Ragionamento geometrico, Lettura attenta, Stima Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Math competition: at least 2 contestants solved exactly 5 problems each

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac{2}{5}$ of the contestants. Moreover, no contestant solved all the $6$ problems. Prove that there are at least $2$ contestants who solved exactly $5$ problems each.

src_imho_2005__Q06