IMO 1992 (33rd IMO, Moscow)

Find a,b,c with (a-1)(b-1)(c-1) dividing abc-1

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 Metodo: Analisi per casi, congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find a,b,c with (a-1) (b-1) (c-1) dividing abc-1

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_imo_1992__Q01

Find all f with f(x^2+f(y))=y+f(x)^2

Let R denote the set of all real numbers. Find all functions f : R →R such that f ³ x2 + f(y) ´ = y + (f(x))2 for all x, y ∈R.

Topic: successioni Abilita: generalizzazione Area: Algebra e Analisi Fonte: apri PDF p.1

Find all f with f(x^2+f(y))=y+f(x)^2

Let R denotes the set of all real numbers. Find all functions f: R →R such that f 3 x2 + f(y) ’ = y + (f(x))2 for all x, y ∈R.

src_imo_1992__Q02

Smallest n of colored edges forcing monochromatic triangle

Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored 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. 33rd International Mathematical Olympiad Second Day - Moscow - July 15, 1992 Time Limit: 41 2 hours

Topic: Combinatoria Metodo: Principio di estremalita, Teoria dei grafi Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Smallest n of colored edges forcing monochromatic triangle

Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored 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. 33rd International Mathematical Olympiad Second Day - Moscow - July 15, 1992 Time limit: 41 2 hours

src_imo_1992__Q03

Find locus of P with incircle tangent and midpoint condition

Nel piano sia $C$ un cerchio, $l$ una retta tangente al cerchio $C$, e $M$ un punto su $l$. Trovare il luogo geometrico di tutti i punti $P$ con la seguente proprietà: esistono due punti $Q$, $R$ su $l$ tali che $M$ è il punto medio di $QR$ e $C$ è il cerchio inscritto nel triangolo $PQR$.

Topic: Geometria piana Metodo: Metodo delle coordinate Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Find the locus of P with incircle tangent and midpoint condition

In the plane $C$ a circle, $l$ a straight tangent to the circle $C$, and $M$ a point on $l$. Find the geometric location of all $P$ points with the following property: there are two points $Q$, $R$ on $l$ such that $M$ is the middle point of $QR$ and $C$ is the circle inscribed in the $PQR$ triangle.

src_imo_1992__Q04

Bound point-set size by product of projection sizes

Sia $S$ un insieme finito di punti nello spazio tridimensionale. Siano $S_x$, $S_y$, $S_z$ gli insiemi costituiti dalle proiezioni ortogonali dei punti di $S$ sul piano $yz$, sul piano $xz$ e sul piano $xy$, rispettivamente. Dimostrare che $|S{|}^2\le |S_x|\cdot |S_y|\cdot |S_z|$ dove $|A|$ denota il numero di elementi dell’insieme finito $A$.

Topic: Combinatoria, Geometria solida Metodo: Disuguaglianze classiche, Doppio conteggio Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Bound point-set size by product of projection sizes

Whether $S$ a finite set of points in three-dimensional space. The sets consisting of the orthogonal projections of the points of $S$ on the plane $yz$, $xz$ and $xy$ are $S_x$, $S_y$, $S_z$ respectively. Show that $|S{|}^2\le |S_x|\cdot |S_y|\cdot |S_z|$ where $|A|$ denotes the number of elements of the finite set $A$.

src_imo_1992__Q05

n^2 as sum of k positive squares, bounds

Per ogni intero positivo $n$, si definisce $S(n)$ come il massimo intero tale che, per ogni intero positivo $k$ con $1\le k\le S(n)$, $n^2$ può essere scritto come somma di $k$ quadrati di interi positivi.

(a) Dimostrare che $S(n)\le n^2-14$ per ogni $n\ge 4$.

(b) Trovare un valore specifico di $n$ per cui $S(n)=n^2-14$.

Topic: Teoria dei Numeri Metodo: Induzione Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

n^2 as sum of k positive squares, bounds

For each positive integer $n$, $S(n)$ is defined as the maximum integer such that, for each positive integer $k$ with $1\le k\le S(n)$, $n^2$ can be written as the sum of $k$ squares of positive integers.

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

(b) Find a specific value of $n$ for which $S(n)=n^2-14$.

src_imo_1992__Q06