IMO 1992 (XXXIII International Mathematical Olympiad)

Find integers a,b,c with divisibility condition

Find all integers $a,b,c$ with $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.

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

Find integers a,b,c with divisibility condition

Find all integers $a,b,c$ with $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.

src_imho_1992__Q01

Find all functions satisfying a functional equation

Let $\mathbf{R}$ denote the set of all real numbers. Find all functions $f:\mathbf{R}\to \mathbf{R}$ such that $f\left(x^2+f(y)\right)=y+{\left(f(x)\right)}^2\text{for all }x,y\in \mathbf{R}.$

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

Find all functions satisfying a functional equation

Let $\mathbf{R}$ denotes the set of all real numbers. Find all functions $f:\mathbf{R}\to \mathbf{R}$ such that $f\left(x^2+f(y)\right)=y+{\left(f(x)\right)}^2\text{for all }x,y\in \mathbf{R}.$

src_imho_1992__Q02

Nine coplanar points, colored edges, monochromatic triangle

Consider nine points in space, no four of which are coplanar. Each pair of points is joined by a line segment which is colored either blue or red or left uncolored. Find the smallest value of $n$ such that whenever exactly $n$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

Topic: Combinatoria Metodo: Estremalità, Casework, Colorazione, Grafi Abilita: Conteggio sistematico, Casework accurato, Ragionamento geometrico, Stima Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Nine coplanar points, colored edges, monochromatic triangle

Consider nine points in space, no four of which are coplanar. Each pair of points is joined by a line segment which is colored either blue or red or left uncolored. Find the smallest value of $n$ such that whenever exactly $n$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

src_imho_1992__Q03

Locus of midpoint on circumscribed circle configuration

In the plane let $C$ be a circle, $L$ a line tangent to the circle $C$, and $M$ a point on $L$. Find the locus of all points $P$ with the following property: there exist two points $Q,R$ on $L$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

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

Locus of midpoint on circumscribed circle configuration

In the plane let $C$ be a circle, $L$ a line tangent to the circle $C$, and $M$ a point on $L$. Find the locus of all points $P$ with the following property: there exist two points $Q,R$ on $L$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

src_imho_1992__Q04

Orthogonal projections of a finite set onto coordinate planes

Let $S$ be a finite set of points in three-dimensional space. Let $S_x$, $S_y$, $S_z$ be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $|S{|}^2\le |S_x|\cdot |S_y|\cdot |S_z|,$ where $|A|$ denotes the number of elements in the finite set $|A|$. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)

Topic: Combinatoria, Algebra Metodo: Disuguaglianze, Doppio conteggio Abilita: Manipolazione algebrica, Astrazione, Modellizzazione, Lettura attenta Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Orthogonal projections of a finite set on coordinate planes

Let $S$ be a finite set of points in three-dimensional space. Let $S_x$, $S_y$, $S_z$ be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $|S{|}^2\le |S_x|\cdot |S_y|\cdot |S_z|,$ where $|A|$ denotes the number of elements in the finite set $|A|$. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)

src_imho_1992__Q05

S(n) greatest integer for sum of k positive squares

For each positive integer $n$, $S(n)$ is defined to be the greatest integer such that, for every positive integer $k\le S(n)$, $n^2$ can be written as the sum of $k$ positive squares.

(a) Prove that $S(n)\le n^2-14$ for each $n\ge 4$.

(b) Find an integer $n$ such that $S(n)=n^2-14$.

(c) Prove that there are infinitely many integers $n$ such that $S(n)=n^2-14$.

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

S(n) greatest integer for sum of k positive squares

For every positive integer $n$, $S(n)$ is defined to be the greatest integer such that, for every positive integer $k\le S(n)$, $n^2$ can be written as the sum of $k$ positive squares.

Prove that for each $n\ge 4$.

**(b) ** Find an integer $n$ such that $S(n)=n^2-14$.

Prove that there are infinitely many integers such that $S(n)=n^2-14$.

src_imho_1992__Q06