IMO 2018 (LIX International Mathematical Olympiad)

Circumcircle, perpendicular bisectors, parallel lines in triangle

Let $\Gamma$ be the circumcircle of acute-angled 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 the minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$, respectively. Prove that the lines $DE$ and $FG$ are parallel (or are the same line).

Topic: Geometria piana Metodo: Simmetria Abilita: Ragionamento geometrico, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Circumcircle, perpendicular bisectors, parallel lines in triangle

Let $\Gamma$ be the circumcircle of acute-angled 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 the minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$, respectively. Prove that the lines $DE$ and $FG$ are parallel (or are the same line).

src_imho_2018__Q01

Find all integers n≥3 with a specific recurrence on real sequences

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

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

Find all integers n≥3 with a specific recurrence on real sequences

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

src_imho_2018__Q02

Existence of anti-Pascal triangle with 2018 rows covering 1 to 1+2+…+2018

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. The following array is an anti-Pascal triangle with four rows which contains every integer from 1 to 10: $\begin{array}{ccccccc}& & & 4& & & \\ & & 2& & 6& & \\ & 5& & 7& & 1& \\ 8& & 3& & 10& & 9\end{array}$ Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to $1+2+\cdots +2018$?

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

Existence of anti-Pascal triangle with 2018 rows covering 1 to 1+2+…+2018

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. The following array is an anti-Pascal triangle with four rows which contains every integer from 1 to 10: $\begin{array}{ccccccc}& & & 4& & & \\ & & 2& & 6& & \\ & 5& & 7& & 1& \\ 8& & 3& & 10& & 9\end{array}$ Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to $1+2+\cdots +2018$?

src_imho_2018__Q03

Game on grid sites: find greatest K Amy can guarantee K red stones

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 her turn, 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 his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone cannot be placed at any distance from any other 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.

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

Game on grid sites: find greatest K Amy can guarantee K red stones

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 her turn, 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 his turn, Ben places a new blue stone on any unoccupied 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.

src_imho_2018__Q04

Infinite sequence of positive integers: prove a_m = a_{m+1} for some M

Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. Suppose that there is an integer $N>1$ such that, for each $n\ge N$, the number $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m=a_{m+1}$ for all $m\ge M$.

Topic: Teoria dei Numeri, Algebra Metodo: Induzione, Telescoping, Invarianti Abilita: Manipolazione algebrica, Ragionamento geometrico, Astrazione, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Infinite sequence of positive integers: tests a_m = a_{m+1} for some M

Let$a_1,a_2,\dots$be an infinite sequence of positive integers. Suppose that there is an integer $N>1$ such that, for each $n\ge N$, the number $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m=a_{m+1}$ for all $m\ge M$.

src_imho_2018__Q05

Convex quadrilateral with angle conditions implies angle sum 180°

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 $\angle BXA+\angle DXC=180^{\circ }$.

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

Convex quadrilateral with angle conditions implies angle sum 180°

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 $\angle BXA+\angle DXC=180^{\circ }$.

src_imho_2018__Q06