IMO 2022 papers

Coppie (n,k) monete con n a sinistra dello stesso tipo

The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k⩽2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k$-th coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering AABBBABA would be $\text{AABBBABA}\to \text{BBBAAABA}\to \text{AAABBBBA}\to \text{BBBBAAAA}\to \text{BBBBAAAA}\to \cdots$ Find all pairs $(n,k)$ with $1⩽k⩽2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

Topic: Combinatoria, Logica, giochi, strategie Metodo: monovarianti Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Pairs (n,k) of coins with n left of the same type

The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k⩽2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k$-th coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering AABBBABA would be $\text{AABBBABA}\to \text{BBBAAABA}\to \text{AAABBBBA}\to \text{BBBBAAAA}\to \text{BBBBAAAA}\to \cdots$ Find all pairs $(n,k)$ with $1⩽k⩽2n$ such that for every initial ordering, at some point during the process, the leftmost $n$ coins will all be of the same type.

src_imo_2022__Q01

R+->R+ con unica y soddisfacente

Let ${\mathbb{R}}^{+}$ denote the set of positive real numbers. Find all functions $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that for each $x\in {\mathbb{R}}^{+}$, there is exactly one $y\in {\mathbb{R}}^{+}$ satisfying $xf(y)+yf(x)⩽2.$

Topic: successioni Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.1

R+->R+ with only satisfactory y

Let ${\mathbb{R}}^{+}$ denotes the set of positive real numbers. Find all functions $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that for each $x\in {\mathbb{R}}^{+}$, there is exactly one $y\in {\mathbb{R}}^{+}$ satisfying $xf(y)+yf(x)⩽2.$

src_imo_2022__Q02

Unico modo di disporre primi con prodotto x^2+x+k

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbours is of the form $x^2+x+k$ for some positive integer $x$.

Topic: Teoria dei Numeri, Combinatoria Metodo: congruenze Abilita: Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Only one way to arrange firsts with product x^2+x+k

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

src_imo_2022__Q03

P,S,Q,R conciclici nel pentagono convesso

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD$, $TC=TE$ and $\angle ABT=\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

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

P,S,Q,R conical in the convex pentagon

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD$, $TC=TE$ and $\angle ABT=\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assumes that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assumes that the points $R,E,A,S$ occur on their line in that order. Prove that the points lie on a circle.

src_imo_2022__Q04

Triple (a,b,p) con a^p=b!+p, p primo

Find all triples $(a,b,p)$ of positive integers with $p$ prime and $a^p=b!+p.$

Topic: Teoria dei Numeri Metodo: congruenze Abilita: Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

Triple (a,b,p) with a^p=b!+p, p first

Find all triples $(a,b,p)$ of positive integers with $p$ prime and $a^p=b!+p.$

src_imo_2022__Q05

Minimo numero di cammini uphill in quadrato nordico

Let $n$ be a positive integer. A Nordic square is an $n\times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

Topic: Combinatoria Metodo: Principio di estremalita, Teoria dei grafi Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Minimum number of northern square uphill walks

Let $n$ be a positive integer. A Nordic square is a $n\times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

src_imo_2022__Q06