IMO 1999 (XL International Mathematical Olympiad)

Finite point sets symmetric under all perpendicular bisectors

Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition:

for any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an axis of symmetry for $S$.

Topic: Geometria piana, Combinatoria Metodo: Simmetria, Estremalità Abilita: Ragionamento geometrico, Astrazione, Lettura attenta Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Finite point sets symmetric under all perpendicular bisectors

Determine the finite sets $S$ of at least three points in the plane which satisfy the following condition:

For any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an axis of symmetry for $S$.

src_imho_1999__Q01

Least constant for symmetric sum inequality

Let $n$ be a fixed integer, with $n\ge 2$.

(a) Determine the least constant $C$ such that the inequality ${\sum }_{1\le i<j\le n}{x}_i{x}_j(x_i^2+x_j^2)\le C{\left({\sum }_{1\le i\le n}{x}_i\right)}^4$ holds for all real numbers $x_1,\cdots ,x_n\ge 0$.

(b) For this constant $C$, determine when equality holds.

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

Least constant for symmetric sum inequality

Let $n$ be a fixed integer, with $n\ge 2$.

(a) Determine the least constant $C$ such that the inequality ${\sum }_{1\le i<j\le n}{x}_i{x}_j(x_i^2+x_j^2)\le C{\left({\sum }_{1\le i\le n}{x}_i\right)}^4$ holds for all real numbers $x_1,\cdots ,x_n\ge 0$.

(b) For this constant $C$, determine when equality holds.

src_imho_1999__Q02

Minimum marked squares so every square is adjacent to one

Consider an $n\times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board are adjacent if they have a common side.

$N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.

Determine the smallest possible value of $N$.

Topic: Combinatoria Metodo: Estremalità, Casework, Doppio conteggio Abilita: Conteggio sistematico, Modellizzazione, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Minimum marked squares so every square is adjacent to one

Consider a $n\times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board are adjacent if they have a common side.

$N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.

Determine the smallest possible value of $N$.

src_imho_1999__Q03

Pairs of positive integers with divisibility condition involving prime

Determine all pairs $(n,p)$ of positive integers such that

$p$ is a prime, $n$ not exceeded $2p$, and $(p-1{)}^n+1$ is divisible by $n^{p-1}$.

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

Pairs of positive integers with divisibility condition involving prime

Determine the pairs $(n,p)$ of positive integers such that

$p$ is a prime, $n$ not exceeded $2p$, and $(p-1{)}^n+1$ is divisible by $n^{p-1}$.

src_imho_1999__Q04

Tangent line CD to circle G2 via intersecting circles configuration

Two circles $G_1$ and $G_2$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_1$ passes through the center of $G_2$. The line passing through the two points of intersection of $G_1$ and $G_2$ meets $G$ at $A$ and $B$. The lines $MA$ and $MB$ meet $G_1$ at $C$ and $D$, respectively.

Prove that $CD$ is tangent to $G_2$.

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

Tangent line CD to circle G2 via intersecting circles configuration

Two circles $G_1$ and $G_2$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_1$ passes through the center of $G_2$. The line passing through the two points of intersection of $G_1$ and $G_2$ meets $G$ at $A$ and $B$. The lines $MA$ and $MB$ meet $G_1$ at $C$ and $D$, respectively.

Prove that $CD$ is tangent to $G_2$.

src_imho_1999__Q05

Find all real functions satisfying given functional equation

Determine all functions $f:\mathbf{R}⟶\mathbf{R}$ such that $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ for all real numbers $x,y$.

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

Find the real functions satisfying given functional equation

Determine all functions $f:\mathbf{R}⟶\mathbf{R}$ such that $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ for all real numbers $x,y$.

src_imho_1999__Q06