IMO 2026 (LXVII International Mathematical Olympiad)

Blackboard of 2026 integers: gcd and lcm/gcd moves leave a unique M>1

There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m>1$ and $n>1$ from different places on the blackboard and replaces these two integers with $\gcd (m,n)\text{and}\frac{\mathrm{lcm}(m,n)}{\gcd (m,n)}.$ He continues to make moves while it is possible to do so.

(a) Prove that, regardless of the choices of Confucius, after finitely many moves, exactly one integer $M$ on the blackboard is greater than $1$.

(b) Prove that the value of $M$ does not depend on the choices of Confucius.

(Note that $\gcd (x,y)$ denotes the greatest common divisor of positive integers $x$ and $y$, and $\mathrm{lcm}(x,y)$ denotes the least common multiple of $x$ and $y$.)

Topic: Teoria dei Numeri Metodo: monovarianti, Fattorizzazione Abilita: Riconoscimento di pattern, Astrazione Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Blackboard of 2026 integers: gcd and lcm/gcd moves leave a unique M>1

There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m>1$ and $n>1$ from different places on the blackboard and replaces these two integers with $\gcd (m,n)\text{and}\frac{\mathrm{lcm}(m,n)}{\gcd (m,n)}.$ He continues to make moves while it is possible to do so.

(a) Prove that, regardless of the choices of Confucius, after finitely many moves, exactly one integer $M$ on the blackboard is greater than $1$.

(b) Prove that the value of $M$ does not depend on the choices of Confucius.

(Note that $\gcd (x,y)$ denotes the greatest common divisor of positive integers $x$ and $y$, and $\mathrm{lcm}(x,y)$ denotes the least common multiple of $x$ and $y$.)

src_imho_2026__Q01

Circumcentre of AKL equidistant from midpoints M and N

Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that $\angle KBA=\angle ACL,\angle LBK=\angle LNC,\text{and}\angle LCK=\angle BMK.$ Let $O$ be the circumcentre of triangle $AKL$. Prove that $OM=ON$.

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

Circumcentre of AKL equidistant from midpoints M and N

Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that $\angle KBA=\angle ACL,\angle LBK=\angle LNC,\text{and}\angle LCK=\angle BMK.$ Let $O$ be the circumcentre of triangle $AKL$. Prove that $OM=ON$.

src_imho_2026__Q02

Liu and Xiang Yu cut a stick of length 1; largest c Liu can guarantee

Let $n$ be a positive integer. Liu Bang and Xiang Yu have a stick of length $1$ and want to divide it between themselves. Liu marks at most $n$ points on the stick, and then Xiang marks at most $n$ points on the stick. The marked points are distinct. Then, the stick is cut at all marked points, creating a number of pieces. Afterwards, they take turns claiming any unclaimed piece of the stick, with Liu going first. Each player’s goal is to maximise the total length of their own pieces.

For each $n$, determine the largest value $c$ such that Liu may guarantee a total length of at least $c$, regardless of Xiang’s play.

Topic: Combinatoria, Disuguaglianze Metodo: Casework, monovarianti Abilita: Astrazione, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Liu and Xiang Yu cut a stick of length 1; largest c Liu can guarantee

Let $n$ be a positive integer. Liu Bang and Xiang Yu have a stick of length $1$ and want to divide it between themselves. Liu marks at most $n$ points on the stick, and then Xiang marks at most $n$ points on the stick. The marked points are distinct. Then, the stick is cut at all marked points, creating a number of pieces. Afterwards, they take turns claiming any unclaimed piece of the stick, with Liu going first. Each player’s goal is to maximise the total length of their own pieces.

For each $n$, determine the largest value $c$ such that Liu may guarantee a total length of at least $c$, regardless of Xiang’s play.

src_imho_2026__Q03

Mulan cutting paper triangles until an angle equals theta

Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^{\circ }<\theta <180^{\circ }$ known to both players. Initially, Shan-Yu makes a paper triangle $T$ with measurements of his choice. Then, they repeatedly perform the following steps:

• If $T$ has at least one angle measuring exactly $\theta$, then the game stops and Mulan wins.
• Otherwise, Mulan chooses a point $P$ on the perimeter of $T$, different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $T$, splitting it into two triangles.
• Shan-Yu discards one of the two triangles. The remaining triangle becomes the new $T$.

For which real values of $\theta$ can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?

Topic: Geometria piana, Combinatoria Metodo: monovarianti, Casework Abilita: Ragionamento geometrico, Astrazione Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Mulan cutting paper triangles until an angle equals theta

Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^{\circ }<\theta <180^{\circ }$ known to both players. Initially, Shan-Yu makes a paper triangle $T$ with measurements of his choice. Then, they repeatedly perform the following steps:

• If $T$ has at least one angle measuring exactly $\theta$, then the game stops and Mulan wins.
• Otherwise, Mulan chooses a point $P$ on the perimeter of $T$, different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $T$, splitting it into two triangles.
• Shan-Yu discards one of the two triangles. The remaining triangle becomes the new $T$.

For which real values of $\theta$ can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?

src_imho_2026__Q04

Positive functions satisfying a two-sided quadratic mean inequality

Let ${\mathbb{R}}_{>0}$ be the set of positive real numbers. Determine all functions $f:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0}$ such that $\frac{x^2+f(y{)}^2}{2}\ge \frac{f(x)+y}{2}\ge \sqrt{xf(y)}$ for every $x,y\in {\mathbb{R}}_{>0}$.

Topic: successioni, Disuguaglianze Metodo: Sostituzione, Disuguaglianze Abilita: Manipolazione algebrica, Astrazione Area: Algebra e Analisi Fonte: apri PDF p.2

Positive functions satisfying a two-sided quadratic mean inequality

Let ${\mathbb{R}}_{>0}$ be the set of positive real numbers. Determine all functions $f:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0}$ such that $\frac{x^2+f(y{)}^2}{2}\ge \frac{f(x)+y}{2}\ge \sqrt{xf(y)}$ for every $x,y\in {\mathbb{R}}_{>0}$.

src_imho_2026__Q05

Sequence of integers >1 sharing a factor with all predecessors is eventually arithmetic

Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers greater than $1$. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd (a_{n+1},a_i)>1$ for every $i=1,2,\dots ,n$. Prove that there exist positive integers $T$ and $L$ such that $a_{n+T}=a_n+L$ for every positive integer $n$.

Topic: Teoria dei Numeri, successioni Metodo: monovarianti, Fattorizzazione Abilita: Riconoscimento di pattern, Astrazione Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

Sequence of integers >1 sharing a factor with all predecessors is eventually arithmetic

Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers greater than $1$. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd (a_{n+1},a_i)>1$ for every $i=1,2,\dots ,n$. Prove that there exist positive integers $T$ and $L$ such that $a_{n+T}=a_n+L$ for every positive integer $n$.

src_imho_2026__Q06