EGMO 2026

A $2026 \times 2026$ board is said to be bordeaux if at least one of its $2026^2$ unit cells is coloured red. A rectangular region made of cells is oddly-rectangular if it contains an odd number of red cells. Determine the largest positive integer $M$ such that, in every possible $2026 \times 2026$ bordeaux board, there exists an oddly-rectangular region of at least $M$ cells.

Note: A rectangular region has sides that are parallel to the sides of the board and contains all of its interior.

Given a positive integer $n$, Marie plays a game where she starts with the number 1 on a blackboard. As many times as she wants, she can choose an integer $j$ such that $1 \leq j \leq n$ and replace the number $V$ on the blackboard with the number $j \cdot R\left(\frac{V}{j}\right)$. Here, $R(x)$ denotes the nearest integer to $x$; if $x$ is exactly halfway between two consecutive integers, it is rounded up. For example, $R(1.3) = 1$ and $R(1.5) = R(1.8) = 2$.

a) Prove that for each given $n$, there is a positive integer $B$ such that Marie can never end up with a number larger than $B$ on the blackboard.

b) For any given $n$, let $f(n)$ be the maximum number obtainable on the blackboard after finitely many replacements. Show that there exists a positive integer $N$ such that for all $n \geq N$, we have that 2026 divides $f(n)$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$, the following holds:

$$f((f(x) + f(y))^2) = (x + y)f(x + y).$$

Let $1 = a_1 \geq a_2 \geq a_3 \geq \cdots$ be an infinite sequence of real numbers such that $a_n = a_{2n} + a_{2n+1}$ for all positive integers $n$. For $r = 2026^{2026}$, prove that

$$\frac{1}{r} \leq a_r \leq \frac{2}{r+1}.$$

Let $ABC$ be an acute triangle with $AC > AB$. Denote by $\omega$ its circumcircle and by $O$ its circumcentre. Let $K$ be the intersection of the tangents to $\omega$ at $B$ and $C$. Circle $ABK$ intersects line $BC$ again at $Z \neq B$. Let $L$ be the midpoint of $KZ$. Let $X$ be the intersection of lines $KZ$ and $AB$. Let $V$ be the point on circle $ABL$ on the same side of $BC$ as $A$ such that $OV$ is perpendicular to $KZ$. Prove that $LV$ is perpendicular to $CX$.

Let $p$ be a prime number and let $n$ be a positive integer such that $p$ does not divide $n$. Denote by $k$ the number of positive divisors of $n$, and by $1 = d_1 < d_2 < \cdots < d_k = n$ the positive divisors of $n$. For $i = 1, 2, \ldots, k$, let $c_i$ be the number of positive divisors $\ell$ of $d_i^2$ such that $d_i - \ell$ is divisible by $p$. Prove that

$$(p-1)(c_1 + c_2 + \cdots + c_k) \geq k^2.$$