Problem

Source: Czech-Polish-Slovak Junior Match 2024, I-1

Tags: combinatorics



Initially, the numbers $1$ and $2$ are written on the board. A move consists of choosing a positive real number $x$ and replacing $(a,b)$ on the board by $\left(a+\frac{x}{b},b+\frac{x}{a}\right)$. Is it possible to create in finitely many moves a situation where the numbers on the board are $2$ and $3$?