Question 77
Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that, for any set $A$ containing exactly 2024 distinct positive integers, the set
$$
S_A := \{f^{(k)}(x) \mid 1 \le k \le 2024,\ x \in A\}
$$
is also a set containing exactly 2024 distinct positive integers.
*Remark*: $f^{(k)}$ denotes the $k$-th iteration of $f$. In other words, $f^{(1)}(x) := f(x)$ and for all $n \ge 1$, $f^{(n+1)}(x) := f(f^{(n)}(x))$.
$$
S_A := \{f^{(k)}(x) \mid 1 \le k \le 2024,\ x \in A\}
$$
is also a set containing exactly 2024 distinct positive integers.
*Remark*: $f^{(k)}$ denotes the $k$-th iteration of $f$. In other words, $f^{(1)}(x) := f(x)$ and for all $n \ge 1$, $f^{(n+1)}(x) := f(f^{(n)}(x))$.