EGMO 2025

For a positive integer $N$ , let $c_{1}< c_{2}< \dots < c_{m}$ be all the positive integers smaller than $N$ that are coprime to $N$ . Find all $N\geq 3$ such that

$$\gcd (N,c_{i} + c_{i + 1})\neq 1$$

for all $1\leq i\leq m - 1$

Here $\gcd (a,b)$ is the largest positive integer that divides both $a$ and $b$ . Integers $a$ and $b$ are coprime if $\gcd (a,b) = 1$ .

An infinite increasing sequence $a_{1}< a_{2}< a_{3}< \dots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_{n}$ terms of the sequence is equal to $a_{n}$ .

Show that there exists an infinite sequence $b_{1},b_{2},b_{3},\ldots$ of positive integers such that for every central sequence $a_{1},a_{2},a_{3},\ldots$ , there are infinitely many positive integers $n$ with $a_{n} = b_{n}$ .

Let $A B C$ be an acute triangle. Points $B$ , $D$ , $E$ , and $C$ lie on a line in this order and satisfy $B D =$ $D E = E C$ . Let $M$ and $N$ be the midpoints of $A D$ and $A E$ , respectively. Let $H$ be the orthocentre of triangle $A D E$ . Let $P$ and $Q$ be points on lines $B M$ and $C N$ , respectively, such that $D$ , $H$ , $M$ , and $P$ are concyclic and $E$ , $H$ , $N$ , and $Q$ are concyclic. Prove that $P$ , $Q$ , $N$ , and $M$ are concyclic.

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

Let $A B C$ be an acute triangle with incentre $I$ and $A B \neq A C$ . Let lines $B I$ and $C I$ intersect the circumcircle of $A B C$ at $P \neq B$ and $Q \neq C$ , respectively. Consider points $R$ and $S$ such that $A Q R B$ and $A C S P$ are parallelograms (with $A Q \parallel R B$ , $A B \parallel Q R$ , $A C \parallel S P$ , and $A P \parallel C S$ ). Let $T$ be the point of intersection of lines $R B$ and $S C$ . Prove that points $R$ , $S$ , $T$ , and $I$ are concyclic.

Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^{2}$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$ , the maximum number of good cells over all possible starting configurations.

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 1, and the sum of the numbers in each column is equal to 1. Define $r_{i}$ to be the largest value in row $i$ , and let $R = r_{1} + r_{2} + \dots + r_{2025}$ . Similarly, define $c_{i}$ to be the largest value in column $i$ , and let $C = c_{1} + c_{2} + \dots + c_{2025}$ .What is the largest possible value of $\frac{R}{C}$ ?

What is the largest possible value of $\frac{R}{C}$ ?