Question 81
Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C+x)$ or $C+P(x)$, where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x)+Q(x)$ down on the board. No polynomial is ever erased from the board.
Given two sets of real numbers, $A=\{a_{1}, a_{2}, \ldots, a_{n}\}$ and $B=\{b_{1}, b_{2}, \ldots, b_{n}\}$, a polynomial $f(x)$ with real coefficients is $(A, B)$-nice if $f(A)=B$, where $f(A)=\{f\left(a_{i}\right): i=1,2, \ldots, n\}$.
Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A|=|B|$, one can produce an $(A, B)$-nice polynomial in a finite number of steps.
Given two sets of real numbers, $A=\{a_{1}, a_{2}, \ldots, a_{n}\}$ and $B=\{b_{1}, b_{2}, \ldots, b_{n}\}$, a polynomial $f(x)$ with real coefficients is $(A, B)$-nice if $f(A)=B$, where $f(A)=\{f\left(a_{i}\right): i=1,2, \ldots, n\}$.
Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A|=|B|$, one can produce an $(A, B)$-nice polynomial in a finite number of steps.