IMO 1983 (XXIV International Mathematical Olympiad)

Find all functions satisfying two functional equations

Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the conditions: (i) $f(xf(y))=yf(x)$ for all positive $x,y$; (ii) $f(x)\to 0$ as $x\to \infty$.

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

Find all functions satisfying two functional equations

Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the conditions: (i) $f(xf(y))=yf(x)$ for all positive $x,y$; (ii) $f(x)\to 0$ as $x\to \infty$.

src_imho_1983__Q01

Midpoints and tangent lines to two coplanar circles

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1{Q}_1$ and $M_2$ be the midpoint of $P_2{Q}_2$. Prove that $\angle O_{1}A{O}_2=\angle M_{1}A{M}_2$.

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

Midpoints and tangent lines to two coplanar circles

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1{Q}_1$ and $M_2$ be the midpoint of $P_2{Q}_2$. Prove that $\angle O_{1}A{O}_2=\angle M_{1}A{M}_2$.

src_imho_1983__Q02

Show largest integer of form 2bc-b-c with no common divisor

Let $a$, $b$ and $c$ be positive integers, no two of which have a common divisor greater than 1. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x$, $y$ and $z$ are non-negative integers.

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

Show largest integer of form 2bc-b-c with no common divisor

Let $a$, $b$ and $c$ be positive integers, no two of which have a common divisor greater than 1. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x$, $y$ and $z$ are non-negative integers.

src_imho_1983__Q03

Partition set into two subsets avoiding right-angled triangle

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Topic: Combinatoria, Geometria piana Metodo: Casework, Colorazione Abilita: Ragionamento geometrico, Modellizzazione, Lettura attenta Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Partition set into two subsets avoiding right-angled triangles

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

src_imho_1983__Q04

Choose 1983 distinct positive integers in arithmetic progression

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression? Justify your answer.

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

Choose 1983 distinct positive integers in arithmetic progression

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression? Justify your answer.

src_imho_1983__Q05

Prove inequality involving sides of a triangle

Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\ge 0.$ Determine when equality occurs.

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

Prove inequality involving sides of a triangle

Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\ge 0.$ Determine when equality occurs.

src_imho_1983__Q06