Problem 1, 2015

A positive real number \(x\) is written on a board. In one move you are allowed to do the following: if a number \(a\) is already on the board, you may write down one of the numbers \(a+1\) or \(\frac{1}{a}\); and if numbers \(a\) and \(b\) with \(a > b\) are already on the board, you may write down one of the numbers \(a+b\) or \(a-b\).

Prove that after finitely many moves it is possible to have the number \(x^2\) on the board.