IMO 2025 (LXVI International Mathematical Olympiad)

Count sunny lines through lattice points in the plane

A line in the plane is called sunny if it is not parallel to any of the $x$-axis, the $y$-axis, and the line $x+y=0$.

Let $n\ge 2$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:

• for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ is on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.

Topic: Combinatoria, Geometria analitica Metodo: Casework, Conteggio Abilita: Ragionamento geometrico, Conteggio sistematico, Lettura attenta Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Count sunny lines through lattice points in the plane

A line in the plane is called sunny if it is not parallel to any of the $x$-axis, the $y$-axis, and the line $x+y=0$.

Let $n\ge 2$ be a given integer. Determine the nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:

• for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ is on at least one of the lines; and - exactly $k$ of the $n$ lines are sunny.

src_imho_2025__Q01

Tangent line through orthocentre in two-circle configuration

Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $D$, such that points $C$, $M$, $N$ and $D$ are on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E\ne A$. Line $AP$ intersects $\Gamma$ again at $F\ne A$. Let $H$ be the orthocentre of triangle $PAB$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

(The orthocentre of a triangle is the point of intersection of its altitudes.)

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

Tangent line through orthocentre in two-circle configuration

Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $D$, such that points $C$, $M$, $N$ and $D$ are on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E\ne A$. Line $AP$ intersects $\Gamma$ again at $F\ne A$. Let $H$ be the orthocentre of triangle $PAB$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

(The orthocentre of a triangle is the point of intersection of its altitudes.)

src_imho_2025__Q02

Find smallest constant c for bonza functions on positive integers

Let $\mathbb{N}$ denote the set of positive integers. A function $f:\mathbb{N}\to \mathbb{N}$ is said to be bonza if $f(a)\mid b^a-f(b{)}^{f(a)}$ for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\le cn$ for all bonza functions $f$ and all positive integers $n$.

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

Find the smallest constant c for bonza functions on positive integers

Let $\mathbb{N}$ denotes the set of positive integers. A function $f:\mathbb{N}\to \mathbb{N}$ is said to be good if $f(a)\mid b^a-f(b{)}^{f(a)}$ for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\le cn$ for all bonza functions $f$ and all positive integers $n$.

src_imho_2025__Q03

Find first term of sequence defined by three largest proper divisors

A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1,a_2,\dots$ consists of positive integers, each of which has at least three proper divisors. For each $n\ge 3$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Topic: Teoria dei Numeri, Combinatoria Metodo: Casework, Fattorizzazione, Induzione Abilita: Riconoscimento di pattern, Conteggio sistematico, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Find first term of sequence defined by three largest proper divisors

A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1,a_2,\dots$ consists of positive integers, each of which has at least three proper divisors. For each $n\ge 3$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine the possible values of $a_1$.

src_imho_2025__Q04

Inkosulty game: find all lambda for Alice winning strategy

Alice and Bazza are playing the inkosulty game, a two-player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$-th turn of the game (starting with $n=1$) the following happens:

• If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that $x_1+x_2+\cdots +x_n\le \lambda n.$
• If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that $x_1^2+x_2^2+\cdots +x_n^2\le n.$

If a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Topic: Combinatoria, Disuguaglianze, Insiemi e funzioni Metodo: Estremalità, Induzione, Casework Abilita: Manipolazione algebrica, Stima, Astrazione, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Inkosulty game: find all the lambda for Alice winning strategy

Alice and Bazza are playing the inkosulty game, a two-player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$-th turn of the game (starting with $n=1$) the following happens:

• If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that $x_1+x_2+\cdots +x_n\le \lambda n.$ - If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that $x_1^2+x_2^2+\cdots +x_n^2\le n.$

If a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine the values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

src_imho_2025__Q05

Minimum tiles to cover every row and column unit square in 2025x2025 grid

Consider a $2025\times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place such that for each row and each column of the grid there is exactly one unit square that is not covered by any tile.

Topic: Combinatoria Metodo: Conteggio, Casework, Estremalità, Doppio conteggio Abilita: Conteggio sistematico, Modellizzazione, Ragionamento geometrico, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Minimum tiles to cover every row and column unit square in 2025x2025 grid

Consider a $2025\times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place such that for each row and each column of the grid there is exactly one unit square that is not covered by any tile.

src_imho_2025__Q06