IMO 2024 (LXV International Mathematical Olympiad)

Real alpha such that floor sums are multiples of n

Determine all real numbers $\alpha$ such that, for every positive integer $n$, the integer $\lfloor \alpha \rfloor +\lfloor 2\alpha \rfloor +\cdots +\lfloor n\alpha \rfloor$ is a multiple of $n$. (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. For example, $\lfloor -\pi \rfloor =-4$ and $\lfloor 2\rfloor =\lfloor 2.9\rfloor =2$.)

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

Real alpha such that floor sums are multiples of n

Determine to real numbers $\alpha$ such that, for every positive integer $n$, the integer $\lfloor \alpha \rfloor +\lfloor 2\alpha \rfloor +\cdots +\lfloor n\alpha \rfloor$ is a multiple of $n$. (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. For example, $\lfloor -\pi \rfloor =-4$ and $\lfloor 2\rfloor =\lfloor 2.9\rfloor =2$.)

src_imho_2024__Q01

Pairs (a,b) with eventually constant gcd of power expressions

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $y$ and $N$ such that $\gcd (a^n+b,b^n+a)=y$ holds for all integers $n\ge N$. (Note that $\gcd (x,y)$ denotes the greatest common divisor of integers $x$ and $y$.)

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

Pairs (a,b) with eventually constant gcd of power expressions

Determine the pairs $(a,b)$ of positive integers for which there exist positive integers $y$ and $N$ such that $\gcd (a^n+b,b^n+a)=y$ holds for all integers $n\ge N$. (Note that $\gcd (x,y)$ denotes the greatest common divisor of integers $x$ and $y$.)

src_imho_2024__Q02

Sequence where a_n counts prior occurrences; eventual periodicity

Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n>N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1,a_2,\dots ,a_{n-1}$.

Prove that at least one of the sequences $a_1,a_3,a_5,\dots$ and $a_2,a_4,a_6,\dots$ is eventually periodic.

(An infinite sequence $b_1,b_2,b_3,\dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p}=b_m$ for all $m\ge M$.)

Topic: Combinatoria Metodo: Invarianti, Ricorsione, Casework Abilita: Astrazione, Riconoscimento di pattern, Conteggio sistematico, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Sequence where a_n counts prior occurrences; possible periodicity

Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n>N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1,a_2,\dots ,a_{n-1}$.

Prove that at least one of the sequences $a_1,a_3,a_5,\dots$ and $a_2,a_4,a_6,\dots$ is eventually periodic.

(An infinite sequence $b_1,b_2,b_3,\dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p}=b_m$ for all $m\ge M$.)

src_imho_2024__Q03

Triangle incircle tangent lines, angle sum equals 180 degrees

Let $ABC$ be a triangle with $AB<AC<BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ meet the circumcircle of triangle $ABC$ at $P\ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.

Prove that $\angle KIL+\angle YPX=180^{\circ }$.

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

Triangle incircle tangent lines, angle sum equal to 180 degrees

Let $ABC$ be a triangle with $AB<AC<BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ meet the circumcircle of triangle $ABC$ at $P\ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.

Prove that $\angle KIL+\angle YPX=180^{\circ }$.

src_imho_2024__Q04

Monster game on 2024x2025 grid, minimum guaranteed attempts

Turbo the snail plays a game on a board with $2024$ rows and $2025$ columns. There are some monsters in the cells of the board, but he knows that there is at most one monster in each row and at most one monster in each column. Initially, Turbo does not know where any of the monsters are. On each attempt, he chooses to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.

Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n^{\mathrm{th}}$ attempt or earlier, regardless of the locations of the monsters.

Topic: Combinatoria Metodo: Estremalità, Casework, Invarianti Abilita: Astrazione, Modellizzazione, Casework accurato, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Monster game on 2024x2025 grid, minimum guaranteed attempts

Turbo the snail plays a game on a board with rows and columns. There are some monsters in the cells of the board, but he knows that there is at most one monster in each row and at most one monster in each column. Initially, Turbo doesn’t know where any of the monsters are. On each attempt, he chooses to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters don’t move, and Turbo remembers whether or not every cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.

Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n^{\mathrm{th}}$ attempt or earlier, regardless of the locations of the monsters.

src_imho_2024__Q05

Aquasiperiodic functions on Q, bound on fixed-point-like preimage

Let $\mathbb{Q}$ be the set of rational numbers. A function $f:\mathbb{Q}\to \mathbb{Q}$ is called aquasiperiodic if the following property holds: for every $x,y\in \mathbb{Q}$, $f(x+f(y))=f(x)+y\text{or}f(f(x)+y)=x+f(y).$ Show that there exists an integer $c$ such that for any aquasiperiodic function $f$ there are at most $c$ different rational numbers $r$ such that $f(r)=f(-r)$, and find the smallest possible value of $c$.

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

Aquasiperiodic functions on Q, bound on fixed-point-like preimage

Let $\mathbb{Q}$ be the set of rational numbers. A function $f:\mathbb{Q}\to \mathbb{Q}$ is called aquasiperiodic if the following property holds: for every $x,y\in \mathbb{Q}$, $f(x+f(y))=f(x)+y\text{or}f(f(x)+y)=x+f(y).$ Show that there exists an integer $c$ such that for any aquasiperiodic function $f$ there are at most $c$ different rational numbers $r$ such that $f(r)=f(-r)$, and find the smallest possible value of $c$.

src_imho_2024__Q06