Question 53
Find all finite sets $\{a_1,a_2,\ldots,a_k\}$ of integers for which there is an injective function $f:\mathbb R\to\mathbb R$ such that, for every positive integer $n$,
$$
\{f^n(x)-x:x\in\mathbb R\}=\{n+a_1,n+a_2,\ldots,n+a_k\}.
$$
Here $f^n$ is the $n$-fold composition of $f$ with itself; for example $f^3(x)=f(f(f(x)))$.
The range notation on the left is written in the sense used by the source solution; the original has the malformed expression $\{x\in\mathbb R:f^n(x)-x\}$.
$$
\{f^n(x)-x:x\in\mathbb R\}=\{n+a_1,n+a_2,\ldots,n+a_k\}.
$$
Here $f^n$ is the $n$-fold composition of $f$ with itself; for example $f^3(x)=f(f(f(x)))$.
The range notation on the left is written in the sense used by the source solution; the original has the malformed expression $\{x\in\mathbb R:f^n(x)-x\}$.