IMO 2018 papers

DE parallela a FG (assi BD e CE)

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $AD=AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Topic: Geometria piana Metodo: Sfruttamento della simmetria Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

*DE parallel to FG (BD and CE axes) *

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $AD=AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

src_imo_2018__Q01

Interi n con successione ciclica ai·ai+1+1=ai+2

Find all integers $n\ge 3$ for which there exist real numbers $a_1,a_2,\dots ,a_{n+2}$ satisfying $a_{n+1}=a_1$, $a_{n+2}=a_2$ and $a_i{a}_{i+1}+1=a_{i+2}$ for $i=1,2,\dots ,n$.

Topic: successioni, Teoria dei Numeri Metodo: congruenze, Ricorsione Abilita: Manipolazione algebrica Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Interi n with cyclic succession ai·ai+1+1=ai+2

Find all integers $n\ge 3$ for which there exist real numbers $a_1,a_2,\dots ,a_{n+2}$ satisfying $a_{n+1}=a_1$, $a_{n+2}=a_2$ and $a_i{a}_{i+1}+1=a_{i+2}$ for $i=1,2,\dots ,n$.

src_imo_2018__Q02

Triangolo anti-Pascal con 2018 righe

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$: $4$ $26$ $571$ $83109$ Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1+2+\cdots +2018$?

Topic: Combinatoria Metodo: congruenze, Induzione Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Anti-Pascal triangle with 2018 lines

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$: $4$ $26$ $571$ $83109$ Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1+2+\cdots +2018$?

src_imo_2018__Q03

Massimo K pietre rosse di Amy (gioco)

A site is any point $(x,y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to $20$. Initially, each of the $400$ sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On each move, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On each move, Ben places a new blue stone on any unoccupied site. They stop as soon as someone cannot move. Find the greatest $K$ such that Amy can always place at least $K$ red stones, no matter how Ben plays.

Topic: Combinatoria, Geometria analitica Metodo: Principio di estremalita, monovarianti Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.2

Maximum K red stones of Amy (play)

A site is any point $(x,y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to $20$. Initially, each of the $400$ sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On each move, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On each move, Ben places a new blue stone on any unoccupied site. They stop as soon as someone can’t move. Find the greatest $K$ such that Amy can always place at least $K$ red stones, no matter how Ben plays.

src_imo_2018__Q04

Successione interi con somma ciclica intera diventa costante

On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest K such that Amy can ensure that she places at least K red stones, no matter how Ben places his blue stones. Problem 5. Let a1, a2, … be an infinite sequence of positive integers. Suppose that there is an integer N > 1 such that, for each n ≥N, the number a1 a2

• a2 a3
• · · + an−1 an
• an a1 is an integer. Prove that there is a positive integer M such that am = am+1 for all m ≥M. Problem 6. A convex quadrilateral ABCD satisfies AB · CD = BC · DA. Point X lies inside ABCD so that ∠XAB = ∠XCD and ∠XBC = ∠XDA. Prove that ∠BXA + ∠DXC = 180°. Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points English (eng), day 2

Topic: successioni, Teoria dei Numeri Metodo: congruenze, monovarianti Abilita: Manipolazione algebrica Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

Integer succession with whole cyclic sum becomes constant

On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) Find the greatest K such that Amy can ensure that she places at least K red stones, no matter how Ben places his blue stones. Problem five. Let a1, a2, … be an infinite sequence of positive integers. Suppose that there is an integer N > 1 such that, for each n ≥N, the number a1 a2 + a2 a3 + · · + an−1 an + an a1 is an integer. Prove that there is a positive integer M such that am = am+1 for all m ≥M. Problem number six. A convex quadrilateral ABCD satisfies AB · CD = BC · DA. Point X lies inside ABCD so that XAB = XCD and XBC = XDA. Prove that BXA + DXC = 180°. Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points English (eng), day 2

src_imo_2018__Q05

Angolo BXA+DXC=180 in quadrilatero con AB·CD=BC·DA

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC\cdot DA$. Point $X$ lies inside $ABCD$ so that $\angle XAB=\angle XCD\text{and}\angle XBC=\angle XDA.$ Prove that $\angle BXA+\angle DXC=180°$.

Topic: Geometria piana Metodo: Sfruttamento della simmetria, Tecniche trigonometriche Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.2

Angle BXA+DXC=180 in the square with AB·CD=BC·DA

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC\cdot DA$. Point $X$ lies inside $ABCD$ so that $\angle XAB=\angle XCD\text{and}\angle XBC=\angle XDA.$ Prove that $\angle BXA+\angle DXC=180°$.

src_imo_2018__Q06