IMO 2000 (XLI International Mathematical Olympiad)

Tangent line meets chords and rays in equal segments

$AB$ is tangent to the circles $CAMN$ and $NMBD$. $M$ lies between $C$ and $D$ on the line $CD$, and $CD$ is parallel to $AB$. The chords $NA$ and $CM$ meet at $P$; the chords $NB$ and $DM$ meet at $Q$. The rays $CA$ and $DB$ meet at $E$. Prove that $PE=QE$.

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

Tangent line meets chords and rays in equal segments

$AB$ is tangent to the circles $CAMN$ and $NMBD$. $M$ lies between $C$ and $D$ on the line $CD$, and $CD$ is parallel to $AB$. The chords $NA$ and $CM$ meet at $P$; the chords $NB$ and $DM$ meet at $Q$. The rays $CA$ and $DB$ meet at $E$. Prove that$PE=QE$.

src_imho_2000__Q01

Inequality for positive reals with product 1

Let $a,b,c$ be positive reals with product 1. Prove that $\left(a-1+\frac{1}{b}\right)\left(b-1+\frac{1}{c}\right)\left(c-1+\frac{1}{a}\right)\le 1.$

Topic: Disuguaglianze, Algebra Metodo: Disuguaglianze Abilita: Manipolazione algebrica, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality for positive reals with product 1

Let $a,b,c$ be positive reals with product 1. Prove that $\left(a-1+\frac{1}{b}\right)\left(b-1+\frac{1}{c}\right)\left(c-1+\frac{1}{a}\right)\le 1.$

src_imho_2000__Q02

Game of moving points on a line, periodicity question

$k$ is a positive real. $N$ is an integer greater than 1. $N$ points are placed on a line, not all coincident. A move is carried out as follows. Pick any two points $A$ and $B$ which are not at the same location. Replace $B$ by another point $B^{\prime }$ to the right of $B$. Replace $A$ by another point $A^{\prime }$ to the left of $A$ such that $A^{\prime }{B}^{\prime }=kAB$. For what values of $k$ can we move the points arbitrarily far to the right by repeated moves?

Topic: Algebra, Combinatoria Metodo: Invarianti, Casework Abilita: Riconoscimento di pattern, Modellizzazione, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Game of moving points on a line, periodicity question

$k$ is a positive real. $N$ is an integer greater than 1. $N$ points are placed on a line, not by coincidence. A move is carried out as follows. Pick any two points $A$ and $B$ which are not at the same location. Replace $B$ by another point $B^{\prime }$ to the right of $B$. Replace $A$ by another point $A^{\prime }$ to the left of $A$ such that $A^{\prime }{B}^{\prime }=kAB$. For what values of $k$ can we move the points arbitrarily far to the right by repeated moves?

src_imho_2000__Q03

Cards in boxes; sum identifies third box

100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?

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

Cards in boxes; sum identifies third box

100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?

src_imho_2000__Q04

N divisible by just 2000 distinct primes, power of 2 condition

Can we find $N$ divisible by just 2000 different primes, so that $N$ divides $2^N+1$? [N may be divisible by a prime power.]

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

N divisible by just 2000 distinct primes, power of 2 condition

Can we find $N$ divisible by just 2000 different primes, so that $N$ divides $2^N+1$? [N may be divisible by a prime power.]

src_imho_2000__Q05

Altitude feet reflected in angle bisectors lie on incircle

Let $A_1{A}_2{A}_3$ be an acute-angled triangle. The foot of the altitude from $A_i$ is $K_i$, and the incircle touches the side opposite $A_i$ at $L_i$. The line $K_1{K}_2$ is reflected in the line $L_1{L}_2$. Similarly, the line $K_2{K}_3$ is reflected in the line $L_2{L}_3$, and the line $K_3{K}_1$ is reflected in $L_3{L}_1$. Show that the three new lines form a triangle with vertices on the incircle.

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

Altitude feet reflected in angle bisectors lie on incircle

Let $A_1{A}_2{A}_3$ be an acute-angled triangle. The foot of the altitude from $A_i$ is $K_i$, and the incircle touches the opposite side $A_i$ at $L_i$. The line $K_1{K}_2$ is reflected in the line $L_1{L}_2$. Similarly, the line $K_2{K}_3$ is reflected in the line $L_2{L}_3$, and the line $K_3{K}_1$ is reflected in $L_3{L}_1$. Show that the three new lines form a triangle with vertices on the incircle.

src_imho_2000__Q06