MathDB

Can we move from (1,2) to (2,3) on the blackboard?

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

May 29, 2024

combinatorics

Problem Statement

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