BALKAN 2026

A set $S$ of positive real numbers is called Aristotelian if for any $x, y, z \in S$ satisfying $x < y < z$, we have $$\frac{z - x}{y} \in S.$$ Find all integers $n \geq 4$ for which there exists an Aristotelian set with exactly $n$ elements.

Let $n$ be a positive integer. A $2n \times 2n$ board is tiled with $2 \times 1$ and $1 \times 2$ dominoes. To pivot a domino is to select one of its two unit squares and rotate the entire domino by $90^\circ$ clockwise, $90^\circ$ anticlockwise, or $180^\circ$ about the centre of that unit square. Prove that it is always possible to simultaneously pivot every domino such that, after all pivots have been performed, the dominoes still tile the board.

Let $ABCD$ be a parallelogram with $\angle DAB < 90^\circ$ and $AB < AD$. Let $H$ be the orthocentre of $\triangle BCD$ and $H'$ be the reflection of $H$ over line $BD$. Line $AH$ intersects the lines $BD, CD$ and $BC$ at $E, F$ and $G$ respectively. Prove that the circumcircles of $\triangle HEH'$ and $\triangle CFG$ are tangent.

Let $n \geq 2$ be an integer. Initially, the number 1 is written $n$ times on a blackboard. An operation consists of choosing two numbers $a$ and $b$ currently on the blackboard, not both zero, and replacing them with the numbers $$\frac{(a - b)^2}{a + b} \quad \text{and} \quad \frac{4ab}{a + b}.$$ Determine all integers $n$ for which it is possible, after a finite number of operations, for the number $n$ to appear on the blackboard.