IMO 1990 (XXXI International Mathematical Olympiad)

Find EG/EF ratio given AM/AB = t

Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If $\frac{AM}{AB}=t,$ find $\frac{EG}{EF}$ in terms of $t$.

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

Find EG/EF ratio given AM/AB = t

Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If $\frac{AM}{AB}=t,$ find $\frac{EG}{EF}$ in terms of $t$.

src_imho_1990__Q01

Minimum k black points guaranteeing a good coloring

Let $n\ge 3$ and consider a set $E$ of $2n-1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$. Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.

Topic: Combinatoria Metodo: Estremalità, Casework, Principio dei cassetti Abilita: Conteggio sistematico, Ragionamento geometrico, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Minimum k black points guaranteeing a good colouring

Let $n\ge 3$ and consider a set $E$ of $2n-1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is “good’ if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$. Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.

src_imho_1990__Q02

Find all integers n > 1 such that (2^n+1)/n^2 is integer

Determine all integers $n>1$ such that $\frac{2^n+1}{n^2}$ is an integer.

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

Find all integers n > 1 such that (2^n+1)/n^2 is integer

Determine all integers $n>1$ such that $\frac{2^n+1}{n^2}$ is an integer.

src_imho_1990__Q03

Construct f: Q+ → Q+ satisfying f(xf(y)) = f(x)/y

Let ${\mathbb{Q}}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb{Q}}^{+}\to {\mathbb{Q}}^{+}$ such that $f(xf(y))=\frac{f(x)}{y}$ for all $x$, $y$ in ${\mathbb{Q}}^{+}$.

Topic: Equazioni funzionali, Insiemi e funzioni Metodo: Backward, Ricorsione Abilita: Manipolazione algebrica, Astrazione, Modellizzazione Area: Algebra e Analisi Fonte: apri PDF p.1

The following table shows the results of the evaluation:

Let${\mathbb{Q}}^{+}$be the set of positive rational numbers. Construct a function $f:{\mathbb{Q}}^{+}\to {\mathbb{Q}}^{+}$ such that $f(xf(y))=\frac{f(x)}{y}$ for all $x$, $y$ in ${\mathbb{Q}}^{+}$.

src_imho_1990__Q04

Two-player integer sequence game; determine winning strategies

Given an initial integer $n_0>1$, two players $\mathcal{A}$ and $\mathcal{B}$ choose integers $n_1$, $n_2$, $n_3$, $\dots$ alternately according to the following rules: Knowing $n_{2k}$, $\mathcal{A}$ chooses any integer $n_{2k+1}$ such that $n_{2k}\le n_{2k+1}\le n_{2k}^2.$ Knowing $n_{2k+1}$, $\mathcal{B}$ chooses any integer $n_{2k+2}$ such that $\frac{n_{2k+1}}{n_{2k+2}}$ is a prime raised to a positive integer power. Player $\mathcal{A}$ wins the game by choosing the number 1990; player $\mathcal{B}$ wins by choosing the number 1. For which $n_0$ does: (a) $\mathcal{A}$ have a winning strategy? (b) $\mathcal{B}$ have a winning strategy? (c) Neither player have a winning strategy?

Topic: Combinatoria, Teoria dei Numeri Metodo: Invarianti, Casework, Backward Abilita: Ragionamento geometrico, Astrazione, Modellizzazione, Casework accurato Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Two player integer sequence game; determine winning strategies

Given an initial integer $n_0>1$, two players $\mathcal{A}$ and $\mathcal{B}$ choose integers $n_1$, $n_2$, $n_3$, $\dots$ alternately according to the following rules: Knowing $n_{2k}$, $\mathcal{A}$ chooses any integer $n_{2k+1}$ such that $n_{2k}\le n_{2k+1}\le n_{2k}^2.$ Knowing $n_{2k+1}$, $\mathcal{B}$ chooses any integer $n_{2k+2}$ such that $\frac{n_{2k+1}}{n_{2k+2}}$ is a prime raised to a positive integer power. Player $\mathcal{A}$ wins the game by choosing the number 1990; player $\mathcal{B}$ wins by choosing the number 1. For which $n_0$ does: (a) $\mathcal{A}$ have a winning strategy? (b) $\mathcal{B}$ have a winning strategy? (c) Neither player has a winning strategy?

src_imho_1990__Q05

Convex 1990-gon with all angles equal and sides 1^2,2^2,…,1990^2

Prove that there exists a convex 1990-gon with the following two properties: (a) All angles are equal. (b) The lengths of the 1990 sides are the numbers $1^2$, $2^2$, $3^2$, $\dots$, $1990^2$ in some order.

Topic: Geometria piana, Combinatoria Metodo: method_casework, Simmetria Abilita: Modellizzazione, Ragionamento geometrico, Astrazione Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Convex 1990-gon with all angles equal and sides 1^2,2^2,…,1990

Prove that there exists a convex 1990-gon with the following two properties: (a) All angles are equal. (b) The lengths of the 1990 sides are the numbers $1^2$, $2^2$, $3^2$, $\dots$, $1990^2$ in some order.

src_imho_1990__Q06