MathDB

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)

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, \dots, a_n \}

and

B = \{ b_1, \dots, b_n \}

, a polynomial

f(x)

with real coefficients is

(A,B)

-nice if

f(A) = B

, where

f(A) = \{ f(a_i) : i = 1, 2, \dots, 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.
Proposed by Navid Safaei, Iran

Problem Statement