IMO 1997 (XXXVIII International Mathematical Olympiad)

Coloring unit-square vertices, chessboard-like function f(m,n)

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_1$ be the total area of the black part and $S_2$ be the total area of the white part. Let $f(m,n)=|S_1-S_2|.$

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

(b) Prove that $f(m,n)\le \frac{1}{2}\max (m,n)$ for all $m$ and $n$.

(c) Show that there is no constant $C$ such that $f(m,n)<C$ for all $m$ and $n$.

Topic: Geometria piana, Combinatoria, Teoria dei Numeri Metodo: Casework, Colorazione Abilita: Ragionamento geometrico, Conteggio sistematico, Manipolazione algebrica Area: Geometria, Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

*Coloring unit-square vertices, chessboard-like function f(m,n) *

In the plane the points with integers are the vertices of unit squares. The squares are alternately colored black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_1$ be the total area of the black part and $S_2$ be the total area of the white part. The value of the underlying assets shall be calculated as follows:

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either even or both odd.

(b) Prove that $f(m,n)\le \frac{1}{2}\max (m,n)$ for all $m$ and $n$.

(c) Show that there is no constant $C$ such that $f(m,n)<C$ for all $m$ and $n$.

src_imho_1997__Q01

Triangle angle bisectors meet circumcircle; prove AU = TB + TC

The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that $AU=TB+TC.$

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

Triangle angle bisectors meet circumcircle; tests AU = TB + TC

The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that $AU=TB+TC.$

src_imho_1997__Q02

Real numbers with bounded partial sums; find permutation with bounded partial sums

Let $x_1,x_2,\dots ,x_n$ be real numbers satisfying the conditions $|x_1+x_2+\cdots +x_n|=1$ and $|x_i|\le \frac{n+1}{2},i=1,2,\dots ,n.$ Show that there exists a permutation $y_1,y_2,\dots ,y_n$ of $x_1,x_2,\dots ,x_n$ such that $|y_1+2y_2+\cdots +n{y}_n|\le \frac{n+1}{2}.$

Topic: Combinatoria, Algebra Metodo: Estremalità, Biiezione, Induzione Abilita: Manipolazione algebrica, Ragionamento geometrico, Astrazione Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Real numbers with bounded partial sums; find permutation with bounded partial sums

Let $x_1,x_2,\dots ,x_n$ be real numbers satisfying the conditions $|x_1+x_2+\cdots +x_n|=1$ and $|x_i|\le \frac{n+1}{2},i=1,2,\dots ,n.$ Show that there exists a permutation $y_1,y_2,\dots ,y_n$ of $x_1,x_2,\dots ,x_n$ such that $|y_1+2y_2+\cdots +n{y}_n|\le \frac{n+1}{2}.$

src_imho_1997__Q03

Silver matrix: n×n matrix with entries from {1,…,2n-1}

An $n\times n$ matrix whose entries come from the set $S=\{1,2,\dots ,2n-1\}$ is called a silver matrix if, for each $i=1,2,\dots ,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no silver matrix for $n=1997$;

(b) silver matrices exist for infinitely many values of $n$.

Topic: Combinatoria, Teoria dei Numeri Metodo: Colorazione, Congruenze, Casework Abilita: Conteggio sistematico, Ragionamento geometrico, Astrazione Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

The value of the underlying assets shall be the sum of the assets of the institution and the liabilities of the institution.

An $n\times n$ matrix whose entries come from the set $S=\{1,2,\dots ,2n-1\}$ is called a silver matrix if, for each $i=1,2,\dots ,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no silver matrix for $n=1997$;

(b) silver matrices exist for infinitely many values of $n$.

src_imho_1997__Q04

Find all integer pairs (a,b) with a,b≥1 satisfying a^(b²)=b^a

Find all pairs $(a,b)$ of integers $a,b\ge 1$ that satisfy the equation $a^{b^2}=b^a.$

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

Find the integer pairs (a,b) with a,b≥1 satisfying a^(b2)=b^a

Find all pairs $(a,b)$ of integers $a,b\ge 1$ that satisfy the equation $a^{b^2}=b^a.$

src_imho_1997__Q05

Count representations of n as sum of powers of 2; prove bounds on f(2^n)

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: $4;2+2;2+1+1;1+1+1+1.$ Prove that, for any integer $n\ge 3$, $2^{n^2/4}<f(2^n)<2^{n^2/2}.$

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

*Count representations of n as sum of powers of 2; prove bounds on f(2^n) *

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For example, $f(4)=4$, because the number $4$ can be represented in the following four ways: $4;2+2;2+1+1;1+1+1+1.$ Prove that, for any integer $n\ge 3$, $2^{n^2/4}<f(2^n)<2^{n^2/2}.$

src_imho_1997__Q06