Question 66
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 $n \gt 100$, 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 $n!$.
$$
a+b,\ a-b,\ b-a,\ a \times b,\ a / b\ (\text{ if } b \neq 0),\ b / a\ (\text{ if } a \neq 0).
$$
Prove that, for every integer $n \gt 100$, 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 $n!$.