IMO 2011 (LII International Mathematical Olympiad)

Maximize value of s4 over four-element subsets of positive integers

Given any set $A=\{a_1,a_2,a_3,a_4\}$ of four distinct positive integers, we denote the sum $a_1+a_2+a_3+a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i,j)$ with $1\le i<j\le 4$ for which $a_i+a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

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

Maximize value of s4 over four-element sub-sets of positive integers

Given any set $A=\{a_1,a_2,a_3,a_4\}$ of four distinct positive integers, we denote the sum $a_1+a_2+a_3+a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i,j)$ with $1\le i<j\le 4$ for which $a_i+a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

src_imho_2011__Q01

Windmill process uses each point as pivot infinitely often

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal{S}$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P\in \mathcal{S}$. The line rotates clockwise about the point $P$ until the first time that the line meets some other point belonging to $\mathcal{S}$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$ until it next meets a point of $\mathcal{S}$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal{S}$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal{S}$ as a pivot infinitely many times.

Topic: Combinatoria, Geometria piana Metodo: Invarianti, Simmetria Abilita: Ragionamento geometrico, Astrazione, Modellizzazione, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

The windmill process uses each point as pivot infinitely often

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assumes that no three points of $\mathcal{S}$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P\in \mathcal{S}$. The line rotates clockwise about the point $P$ until the first time that the line meets some other point belonging to $\mathcal{S}$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$ until it next meets a point of $\mathcal{S}$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal{S}$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal{S}$ as a pivot infinitely many times.

src_imho_2011__Q02

Functional inequality implying f(x)=0 for x<=0

Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies $f(x+y)\le yf(x)+f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x\le 0$.

Topic: Equazioni funzionali, Algebra Metodo: Backward, Estremalità Abilita: Manipolazione algebrica, Lettura attenta, Astrazione Area: Algebra e Analisi Fonte: apri PDF p.1

Functional inequality implying f(x)=0 for x<=0

Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies $f(x+y)\le yf(x)+f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x\le 0$.

src_imho_2011__Q03

Count ways to place weights on balance without right pan heavier

Let $n>0$ be an integer. We are given a balance and $n$ weights of weight $2^0,2^1,\dots ,2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done.

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

Count ways to place weights on balance without right pan heavier

Let $n>0$ be an integer. We are given a balance and $n$ weights of weight $2^0,2^1,\dots ,2^{n-1}$. We are to place each of the weights on the balance, one after the other, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways this can be done.

src_imho_2011__Q04

Divisibility property of f(m)-f(n) by f(m-n) for positive integer function

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m)-f(n)$ is divisible by $f(m-n)$. Prove that, for all integers $m$ and $n$ with $f(m)\le f(n)$, the number $f(n)$ is divisible by $f(m)$.

Topic: Teoria dei Numeri, Algebra Metodo: Congruenze, Fattorizzazione Abilita: Manipolazione algebrica, Lettura attenta, Astrazione Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Divisibility property of f(m) -f(n) by f(m-n) for positive integer function

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m)-f(n)$ is divisible by $f(m-n)$. Prove that, for all integers $m$ and $n$ with $f(m)\le f(n)$, the number $f(n)$ is divisible by $f(m)$.

src_imho_2011__Q05

Reflections of tangent line through vertices meet circumcircle tangentially

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let ${\ell }_a$, ${\ell }_b$, ${\ell }_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$, $AB$, respectively. Show that the circumcircle of the triangle determined by the lines ${\ell }_a$, ${\ell }_b$, ${\ell }_c$ is tangent to the circle $\Gamma$.

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

Reflections of tangent line through vertices meet circumcircle tangentially

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let ${\ell }_a$, ${\ell }_b$, ${\ell }_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$, $AB$, respectively. Show that the circumcircle of the triangle determined by the lines ${\ell }_a$, ${\ell }_b$, ${\ell }_c$ is tangent to the circle $\Gamma$.

src_imho_2011__Q06