RMM 2015 Problem 3

A finite list of rational numbers is written on a blackboard. In an operation, we choose any two numbers $a, b$, erase them, and write down one of the numbers

$$a+b, a-b, b-a, a \times b, a / b(\text { if } b \neq 0), b / a(\text { if } a \neq 0) .$$

Prove that, for every integer $n>100$, there are only finitely many integers $k \geq 0$, such that, starting from the list

$$k+1, k+2, \ldots, k+n$$

it is possible to obtain, after $n-1$ operations, the value $n$ !. (United Kingdom) Alexander Betts