IMO 1994 (XXXV International Mathematical Olympiad)

Distinct elements of {1..n} with pairwise sum condition

Let $m$ and $n$ be positive integers. Let $a_1,a_2,\dots ,a_m$ be distinct elements of $\{1,2,\dots ,n\}$ such that whenever $a_i+a_j\le n$ for some $i,j$, $1\le i\le j\le m$, there exists $k$, $1\le k\le m$, with $a_k=a_i+a_j$. Prove that $a_1+a_2+\cdots +a_m\ge \frac{n+1}{2}.$

Topic: Combinatoria, Teoria dei Numeri Metodo: Estremalità, Induzione Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Distinct elements of {1..n} with pairwise sum condition

Let $m$ and $n$ be positive integers. Let $a_1,a_2,\dots ,a_m$ be distinct elements of $\{1,2,\dots ,n\}$ such that whenever $a_i+a_j\le n$ for some $i,j$, $1\le i\le j\le m$, there exists $k$, $1\le k\le m$, with $a_k=a_i+a_j$. Prove that $a_1+a_2+\cdots +a_m\ge \frac{n+1}{2}.$

src_imho_1994__Q01

Isosceles triangle ABC with AB=AC; perpendicularity condition

$ABC$ is an isosceles triangle with $AB=AC$. Suppose that

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E$, $Q$, $F$ are distinct and collinear. Prove that $OQ$ is perpendicular to $EF$ if and only if $QE=QF$.

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

Isosceles triangle ABC with AB=AC; perpendicularity condition

$ABC$ is an isosceles triangle with $AB=AC$. Suppose that 1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$; 2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$; 3. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E$, $Q$, $F$ are distinct and collinear. Prove that $OQ$ is perpendicular to $EF$ if and only if $QE=QF$.

src_imho_1994__Q02

Binary representations with exactly three 1s; existence of k

For any positive integer $k$, let $f(k)$ be the number of elements in the set $\{k+1,k+2,\dots ,2k\}$ whose base $2$ representation has precisely three $1$s. (a) Prove that, for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k)=m$. (b) Determine all positive integers $m$ for which there exists exactly one $k$ with $f(k)=m$.

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

Binary representations with exactly three 1s; existence of k

For any positive integer $k$, let $f(k)$ be the number of elements in the set $\{k+1,k+2,\dots ,2k\}$ whose base $2$ representation has precisely three $1$s. (a) Prove that, for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k)=m$. (b) Determine the positive integers $m$ for which there exists exactly one $k$ with $f(k)=m$.

src_imho_1994__Q03

Find all ordered pairs (m,n) of positive integers with (n^3+1)/(mn-1) integer

Determine all ordered pairs $(m,n)$ of positive integers such that $\frac{n^3+1}{mn-1}$ is an integer.

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

Find the ordered pairs (m,n) of positive integers with (n^3+1)/(mn-1) integer

Determine to ordered pairs $(m,n)$ of positive integers such that $\frac{n^3+1}{mn-1}$ is an integer.

src_imho_1994__Q04

Functional equation on reals greater than -1

Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f:S\to S$ satisfying the two conditions:

1. $f(x+f(y)+xf(y))=y+f(x)+yf(x)$ for all $x$ and $y$ in $S$;
2. $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$.

Topic: Equazioni funzionali, Algebra Metodo: Casework, Backward Abilita: Manipolazione algebrica, Astrazione, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Functional equation on reals greater than -1

Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f:S\to S$ satisfying the two conditions: 1. $f(x+f(y)+xf(y))=y+f(x)+yf(x)$ for all $x$ and $y$ in $S$; 2. $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$.

src_imho_1994__Q05

Set A of positive integers; product of k-element subsets is a product of k primes

Show that there exists a set $A$ of positive integers with the following property: for any infinite set $S$ of primes there exist two positive integers $m\in A$ and $n\notin A$ each of which is a product of $k$ distinct elements of $S$ for some $k\ge 2$.

Topic: Combinatoria, Teoria dei Numeri, Insiemi e funzioni Metodo: Induzione, Estremalità Abilita: Astrazione, Modellizzazione, Riconoscimento di pattern, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Set A of positive integers; product of k-element subsets is a product of k primes

Show that there exists a set $A$ of positive integers with the following property: for any infinite set $S$ of primes there exist two positive integers $m\in A$ and $n\notin A$ each of which is a product of $k$ distinct elements of $S$ for some $k\ge 2$.

src_imho_1994__Q06