IMO 2023 (LXIV International Mathematical Olympiad)

Composite integers n>1 with divisibility condition on divisors

Determine all composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\dots ,d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\cdots <d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1\le i\le k-2$.

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

Composite integers n>1 with divisibility condition on divisors

Determine the composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\dots ,d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\cdots <d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1\le i\le k-2$.

src_imho_2023__Q01

Circumcircle tangent line meets angle bisector on internal bisector

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ at $E\ne A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P\ne B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

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

Circumcircle tangent line meets angle bisector on internal bisector

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ at $E\ne A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P\ne B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

src_imho_2023__Q02

Infinite sequences of positive integers admitting a polynomial recurrence

For each integer $k\ge 2$, determine all infinite sequences of positive integers $a_1,a_2,\dots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1}{x}^{k-1}+\cdots +c_{1}x+c_0$, where $c_0,c_1,\dots ,c_{k-1}$ are non-negative integers, such that $P(a_n)=a_{n+1}{a}_{n+2}\cdots a_{n+k}$ for every integer $n\ge 1$.

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

Infinite sequences of positive integers admitting a polynomial recurrence

For each integer $k\ge 2$, determine the infinite sequences of positive integers $a_1,a_2,\dots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1}{x}^{k-1}+\cdots +c_{1}x+c_0$, where $c_0,c_1,\dots ,c_{k-1}$ are non-negative integers, such that $P(a_n)=a_{n+1}{a}_{n+2}\cdots a_{n+k}$ for each integer $n\ge 1$.

src_imho_2023__Q03

Sum involving reciprocals is always an integer

Let $x_1,x_2,\dots ,x_{2023}$ be pairwise different positive real numbers such that $a_n=\sqrt{(x_1+x_2+\cdots +x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}\right)}$ is an integer for every $n=1,2,\dots ,2023$. Prove that $a_{2023}\ge 3034$.

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

Sum involving reciprocals is always an integer

Let $x_1,x_2,\dots ,x_{2023}$ be pairwise different positive real numbers such that $a_n=\sqrt{(x_1+x_2+\cdots +x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}\right)}$ is an integer for every $n=1,2,\dots ,2023$. Prove that $a_{2023}\ge 3034$.

src_imho_2023__Q04

Ninja path with k red circles in Japanese triangle

Let $n$ be a positive integer. A Norwegian triangle is a triangle of $1+2+\cdots +n$ circles arranged in an equilateral triangular shape such that for each $i=1,2,\dots ,n$, the $i^{\text{th}}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Norwegian triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Norwegian triangle with $n=6$, along with a ninja path in that triangle containing two red circles.

In terms of $n$, find the greatest $k$ such that in each Norwegian triangle there is a ninja path containing at least $k$ red circles.

Topic: Combinatoria Metodo: Estremalità, Casework, Induzione Abilita: Ragionamento geometrico, Riconoscimento di pattern, Conteggio sistematico, Astrazione Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Ninja path with k red circles in Japanese triangle

Let $n$ be a positive integer. A Norwegian triangle is a triangle of $1+2+\cdots +n$ circles arranged in an equilateral triangular shape such that for each $i=1,2,\dots ,n$, the $i^{\text{th}}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Norwegian triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Norwegian triangle with $n=6$, along with a ninja path in that triangle containing two red circles.

In terms of $n$, find the greatest $k$ such that in each Norwegian triangle there is a ninja path containing at least $k$ red circles.

src_imho_2023__Q05

Interior points of equilateral triangle with angle sum condition

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $B{A}_1=A_{1}C$, $C{B}_1=B_{1}A$, $A{C}_1=C_{1}B$, and $\angle B{A}_{1}C+\angle C{B}_{1}A+\angle A{C}_{1}B=480^{\circ }.$ Let $B{C}_1$ and $C{B}_1$ meet at $A_2$; let $C{A}_1$ and $A{C}_1$ meet at $B_2$; and let $A{B}_1$ and $B{A}_1$ meet at $C_2$. Prove that if the triangle $A_1{B}_1{C}_1$ is scalene, then the three lines $A_1{A}_2$, $B_1{B}_2$, $C_1{C}_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

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

Interior points of equilateral triangle with angle sum condition

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $B{A}_1=A_{1}C$, $C{B}_1=B_{1}A$, $A{C}_1=C_{1}B$, and $\angle B{A}_{1}C+\angle C{B}_{1}A+\angle A{C}_{1}B=480^{\circ }.$ Let $B{C}_1$ and $C{B}_1$ meet at $A_2$; let $C{A}_1$ and $A{C}_1$ meet at $B_2$; and let $A{B}_1$ and $B{A}_1$ meet at $C_2$. Prove that if the triangle $A_1{B}_1{C}_1$ is scalene, then the three lines $A_1{A}_2$, $B_1{B}_2$, $C_1{C}_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

src_imho_2023__Q06