IMO 2019 (LX International Mathematical Olympiad)

Find all functions f:Z->Z satisfying f(2a)+2f(b)=f(f(a+b))

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a$ and $b$, $f(2a)+2f(b)=f(f(a+b)).$

Topic: Algebra, Equazioni funzionali Metodo: Induzione, Simmetria Abilita: Manipolazione algebrica, Lettura attenta, Riconoscimento di pattern Area: Algebra e Analisi Fonte: apri PDF p.1

Find to the functions f:Z->Z satisfying f(2a) +2f(b)=f(f(a+b))

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a$ and $b$, $f(2a)+2f(b)=f(f(a+b)).$

src_imho_2019__Q01

Concyclicity of two points defined via angle and parallel conditions in triangle

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $A{A}_1$ and $B{B}_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $P{B}_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle P{P}_{1}C=\angle BAC$. Let $Q_1$ be a point on line $Q{A}_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle C{Q}_{1}Q=\angle CBA$.

Prove that points $P$, $Q$, $P_1$, and $Q_1$ are concyclic.

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

Concyclicity of two points defined via angle and parallel conditions in triangle

In the $ABC$ triangle, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $A{A}_1$ and $B{B}_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $P{B}_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle P{P}_{1}C=\angle BAC$. Let $Q_1$ be a point on line $Q{A}_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle C{Q}_{1}Q=\angle CBA$.

Prove that points $P$, $Q$, $P_1$, and $Q_1$ are concyclic.

src_imho_2019__Q02

Social network friendship changes; prove at most one friend remains

A social network has 2019 users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:

Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and $A$ is no longer friends with $C$. All other friendship statuses are unchanged.

Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

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

Social network friendship changes; prove at most one friend remains

A social network has 2019 users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:

Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and $A$ is no longer friends with $C$. All other friendship statuses are unchanged.

Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

src_imho_2019__Q03

Find all pairs (k,n) of positive integers satisfying k^2=(2^n-1)(2^(n-1)-1)…(2^(n+1)-1)

Find all pairs $(k,n)$ of positive integers such that $k!=(2^n-1)(2^n-2)(2^n-4)\cdots (2^n-2^{n-1}).$

Topic: Teoria dei Numeri, Combinatoria Metodo: Fattorizzazione, Casework, Congruenze Abilita: Manipolazione algebrica, Lettura attenta, Conteggio sistematico, Stima Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

*Find all pairs (k,n) of positive integers satisfying k^2=(2

Find all pairs $(k,n)$ of positive integers such that $k!=(2^n-1)(2^n-2)(2^n-4)\cdots (2^n-2^{n-1}).$

src_imho_2019__Q04

Bank of Bath coins problem: show process terminates and find average L(C)=0

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k^{\text{th}}$ coin from the left; otherwise, he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT\to HHT\to HTT\to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT)=3$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

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

Bank of Bath coins problem: show process terminates and find average

The Bank of Bath issues coins with a $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k^{\text{th}}$ coin from the left; otherwise, he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT\to HHT\to HTT\to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT)=3$. Determine the average value of $L(C)$ over all possible initial configurations $C$.

src_imho_2019__Q05

Lines DI and PQ meet on altitude through A in acute triangle with incircle tangencies

Let $I$ be the incentre of acute triangle $ABC$ with $AB\ne AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

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

Lines DI and PQ meet on altitude through A in acute triangle with incircle tangencies

Let $I$ be the incentre of acute triangle $ABC$ with $AB\ne AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

src_imho_2019__Q06