IMO 2014 (LV International Mathematical Olympiad)

Infinite sequence of positive integers with a ratio bound

Let $a_0<a_1<a_2<\cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\ge 1$ such that $a_n<\frac{a_0+a_1+\cdots +a_n}{n}\le a_{n+1}.$

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

Infinite sequence of positive integers with a ratio bound

Let$a_0<a_1<a_2<\cdots$be an infinite sequence of positive integers. Prove that there exists a unique integer $n\ge 1$ such that $a_n<\frac{a_0+a_1+\cdots +a_n}{n}\le a_{n+1}.$

src_imho_2014__Q01

Peaceful rook configurations on n×n board with no k×k empty square

Let $n\ge 2$ be an integer. Consider an $n\times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is called peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k\times k$ square which does not contain a rook on any of its $k^2$ unit squares.

Topic: Combinatoria Metodo: Estremalità, Casework, Induzione Abilita: Conteggio sistematico, Riconoscimento di pattern, Casework accurato Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Peaceful rook configurations on n×n board with no k×k empty square

Let $n\ge 2$ be an integer. Consider a $n\times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is called peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k\times k$ square which does not contain a rook on any of its $k^2$ unit squares.

src_imho_2014__Q02

BD tangent to circumcircle of triangle TSH in convex quadrilateral

Convex quadrilateral $ABCD$ has $\angle ABC=\angle CDA=90^{\circ }$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and $\angle CHS-\angle CSB=90^{\circ },\angle THC-\angle DTC=90^{\circ }.$ Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

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

BD tangent to circumcircle of triangle TSH in convex quadrilateral

The convex quadrilateral $ABCD$ has $\angle ABC=\angle CDA=90^{\circ }$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and $\angle CHS-\angle CSB=90^{\circ },\angle THC-\angle DTC=90^{\circ }.$ Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

src_imho_2014__Q03

Lines BM and CN concur on circumcircle of triangle ABC

Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$ so that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$.

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

Lines BM and CN concur on circumcircle of triangle ABC

Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$ so that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$.

src_imho_2014__Q04

Bank of Cape Town coins split into groups of at most 1

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac{1}{n}$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99+\frac{1}{2}$, prove that it is possible to split this collection into at most $100$ groups, such that each group has total value at most $1$.

Topic: Combinatoria, Teoria dei Numeri Metodo: Estremalità, Casework, Induzione Abilita: Modellizzazione, Conteggio sistematico, Casework accurato, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Bank of Cape Town coins split into groups of at most 1

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac{1}{n}$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99+\frac{1}{2}$, prove that it is possible to split this collection into at most $100$ groups, such that each group has total value at most $1$.

src_imho_2014__Q05

Coloring sqrt(n) lines blue with completely blue finite region

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ of the lines blue in such a way that none of its finite regions has a completely blue boundary.

\textit{Note:} Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

Topic: Combinatoria Metodo: Colorazione, Grafi, Estremalità, Induzione Abilita: Modellizzazione, Ragionamento geometrico, Riconoscimento di pattern, Stima Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Coloring sqrt(n) lines blue with completely blue finite region

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ of the lines blue in such a way that none of its finite regions has a completely blue boundary.

\textit{Note:} Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

src_imho_2014__Q06