IMO 1983 (24th International Mathematical Olympiad)

Find all positive functions with f(xf(y))=yf(x) tending to 0

Find all functions $f$ defined on the set of positive real numbers which take positive 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: successioni Abilita: generalizzazione Area: Algebra e Analisi Fonte: apri PDF p.1

Find to positive functions with f(xf(y))=yf(x) tending to 0

Find all functions $f$ defined on the set of positive real numbers which take positive values and satisfy the conditions:

(i) $f(xf(y))=yf(x)$ for the positive $x,y$;

(ii) $f(x)\to 0$ as $x\to \infty$.

src_imo_1983__Q01

Prove angle O1AO2 equals angle M1AM2 for two 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: Tecniche trigonometriche Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Prove angle O1AO2 equals angle M1AM2 for two 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_imo_1983__Q02

Largest integer not representable as nonnegative combination (Frobenius)

Let $a,b,c$ 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 $x,y,z$ are non-negative integers.

Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

*Largest integer not representable as nonnegative combination (Frobenius) *

Let $a,b,c$ 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 $x,y,z$ are non-negative integers.

src_imo_1983__Q03

Two-coloring of equilateral triangle edges yields right 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 disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Topic: Combinatoria, Geometria piana Metodo: Principio dei cassetti, parita Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Two-coloring of equilateral triangle edges yields right 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 disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

src_imo_1983__Q04

Choose 1983 integers with no three-term 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: congruenze Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Choose 1983 integers with no three-term 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_imo_1983__Q05

Prove cyclic triangle-side inequality and find equality case

Let $a,b,c$ 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 Metodo: Disuguaglianze classiche Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.1

Prove cyclic triangle-side inequality and find equality case

Let’s 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_imo_1983__Q06