IMO 2012 (LIII International Mathematical Olympiad)

Incircle tangency and midpoint on triangle bisector

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

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

Incircle tangency and midpoint on triangle bisector

Given triangle $ABC$ the point $J$ is the center of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

src_imho_2012__Q01

Inequality with positive reals summing to n minus 1

Let $n\ge 3$ be an integer, and let $a_2,a_3,\dots ,a_n$ be positive real numbers such that $a_2{a}_3\cdots a_n=1$. Prove that $(1+a_2{)}^2(1+a_3{)}^3\cdots (1+a_n{)}^n>n^n.$

Topic: Disuguaglianze Metodo: Disuguaglianze, Induzione Abilita: Manipolazione algebrica, Stima, Ragionamento geometrico Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality with positive reals summing to n minus 1

Let $n\ge 3$ be an integer, and let $a_2,a_3,\dots ,a_n$ be positive real numbers such that $a_2{a}_3\cdots a_n=1$. Prove that $(1+a_2{)}^2(1+a_3{)}^3\cdots (1+a_n{)}^n>n^n.$

src_imho_2012__Q02

Guessing game: player A specifies positive integer set, B lies at most once

The liar’s guessing game is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players.

At the start of the game $A$ chooses integers $x$ and $N$ with $1\le x\le N$. Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wants. After each question, player $A$ must immediately answer it with yes or no, but is allowed to lie as many times as she wants with the only restriction that, among any $k+1$ consecutive answers, at least one answer must be truthful.

After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins. If $x$ does not belong to $X$, then $A$ wins; otherwise, he loses. Prove that:

1. If $n\ge 2^k$, then $B$ can guarantee a win.
2. For all sufficiently large $k$, there exists an integer $n\ge 1.99^k$ such that $B$ cannot guarantee a win.

Topic: Combinatoria, Logica Metodo: Induzione, Invarianti, Casework Abilita: Ragionamento geometrico, Lettura attenta, Modellizzazione, Casework accurato Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Guessing game: player A specifies positive integer set, B lies at most once

The liar’s guessing game is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players.

At the start of the game $A$ chooses integers $x$ and $N$ with $1\le x\le N$. Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wants. After each question, player $A$ must immediately answer it with yes or no, but is allowed to lie as many times as she wants with the only restriction that, among any $k+1$ consecutive answers, at least one answer must be truthful.

After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins. If $x$ does not belong to $X$, then $A$ wins; otherwise, he loses. Prove that:

1. If $n\ge 2^k$, then $B$ can guarantee a win. 2. For the sufficiently large $k$, there exists an integer $n\ge 1.99^k$ such that $B$ cannot guarantee a win.

src_imho_2012__Q03

Find all functions f: Z to Z satisfying a+b+c=0 functional equation

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: $f(a{)}^2+f(b{)}^2+f(c{)}^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$ (Here $\mathbb{Z}$ denotes the set of integers.)

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

Find all functions f: Z to Z satisfying a+b+c=0 functional equation

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:

src_imho_2012__Q04

Triangle geometry: altitude foot, midpoint, and intersection equality

Let $ABC$ be a triangle with $\angle BCA=90^{\circ }$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

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

Triangle geometry: altitude foot, midpoint, and intersection equality

Let $ABC$ be a triangle with $\angle BCA=90^{\circ }$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that$MK=ML$.

src_imho_2012__Q05

Find all positive integers n with non-negative integer representation as sum of unit fractions

Find all positive integers $n$ for which there exist non-negative integers $a_1,a_2,\dots ,a_n$ such that $\frac{1}{2^{a_1}}+\frac{1}{2^{a_2}}+\cdots +\frac{1}{2^{a_n}}=\frac{1}{3^{a_1}}+\frac{2}{3^{a_2}}+\cdots +\frac{n}{3^{a_n}}=1.$

Topic: Combinatoria, Teoria dei Numeri Metodo: Induzione, Casework, Invarianti Abilita: Manipolazione algebrica, Conteggio sistematico, Riconoscimento di pattern, Stima Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find all positive integers n with non-negative integer representation as sum of unit fractions

Find all positive integers $n$ for which there exist non-negative integers $a_1,a_2,\dots ,a_n$ such that $\frac{1}{2^{a_1}}+\frac{1}{2^{a_2}}+\cdots +\frac{1}{2^{a_n}}=\frac{1}{3^{a_1}}+\frac{2}{3^{a_2}}+\cdots +\frac{n}{3^{a_n}}=1.$

src_imho_2012__Q06