IMO 2002 (XLIII International Mathematical Olympiad)

Coloring pairs in a set, equal type-1 and type-2 subsets

$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h^{\prime }\le h$, $k^{\prime }\le k$, then $(h^{\prime },k^{\prime })$ is also red. A type 1 subset of $S$ has $n$ blue elements with different first members and a type 2 subset of $S$ has $n$ blue elements with different second members. Show that there are the same number of type 1 and type 2 subsets.

Topic: Combinatoria Metodo: Biiezione, Simmetria Abilita: Astrazione, Conteggio sistematico, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Coloring pairs in a set, equal type-1 and type-2 subsets

$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h^{\prime }\le h$, $k^{\prime }\le k$, then $(h^{\prime },k^{\prime })$ is also red. A type 1 subset of $S$ has $n$ blue elements with different first members and a type 2 subset of $S$ has $n$ blue elements with different second members. Show that there are the same number of type 1 and type 2 subsets.

src_imho_2002__Q01

Geometry: incenter lies on chord of circumcircle

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC>60^{\circ }$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $OD$ meets $AC$ at $J$. Show that $J$ is the incentre of triangle $CEF$.

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

Geometry: incenter lies on chord of circumcircle

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC>60^{\circ }$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $OD$ meets $AC$ at $J$. Show that $J$ is the incenter of triangle $CEF$.

src_imho_2002__Q02

Find all pairs of integers m>2, n>2 with k^n-1 divides k^m-1

Find all pairs of integers $m>2$, $n>2$ such that there are infinitely many positive integers $k$ for which $k^n-1$ divides $k^m-1$.

Topic: Teoria dei Numeri Metodo: Fattorizzazione, Congruenze, Induzione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Astrazione Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find all pairs of integers m>2, n>2 with k^n-1 divides k

Find all pairs of integers $m>2$, $n>2$ such that there are infinitely many positive integers $k$ for which $k^n-1$ divides $k^m-1$.

src_imho_2002__Q03

Positive divisors of n>1 in sequence with divisibility condition

The positive divisors of an integer $n>1$ are $d_1<d_2<\cdots <d_k$, so that $d_1=1$, $d_k=n$. Let $d=d_1{d}_2+d_2{d}_3+\cdots +d_{k-1}{d}_k$. Show that $d<n^2$ and find all $n$ for which $d$ divides $n^2$.

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

Positive divisors of n>1 in sequence with divisibility condition

The positive divisors of an integer $n>1$ are $d_1<d_2<\cdots <d_k$, so that $d_1=1$, $d_k=n$. Let $d=d_1{d}_2+d_2{d}_3+\cdots +d_{k-1}{d}_k$. Show that $d<n^2$ and find the $n$ for which $d$ divides $n^2$.

src_imho_2002__Q04

Real functions satisfying (f(x)+f(y))(f(u)+f(v))=f(xu-yv)+f(xv+yu)

Find all real-valued functions on the reals such that $(f(x)+f(y))(f(u)+f(v))=f(xu-yv)+f(xv+yu)$ for all $x,y,u,v$.

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

This is the total number of functions that are satisfying (f(x) +f(y))

Find all real-valued functions on the reals such that $(f(x)+f(y))(f(u)+f(v))=f(xu-yv)+f(xv+yu)$ for all $x,y,u,v$.

src_imho_2002__Q05

n>2 circles of radius 1, no line meets more than two

$n>2$ circles of radius 1 are drawn in the plane so that no line meets more than two of the circles. Their centres are $O_1,O_2,\dots ,O_n$. Show that ${\sum }_{i<j}1/O_i{O}_j\le (n-1)\pi /4$.

Topic: Geometria piana, Combinatoria Metodo: Doppio conteggio, Disuguaglianze, Simmetria Abilita: Stima, Modellizzazione, Ragionamento geometrico, Conteggio sistematico Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

n>2 circles of radius 1, no line meets more than two

$n>2$ circles of radius 1 are drawn in the plane so that no line meets more than two of the circles. Their centres are $O_1,O_2,\dots ,O_n$. Show that${\sum }_{i<j}1/O_i{O}_j\le (n-1)\pi /4$.

src_imho_2002__Q06