MathDB

Let

n \ge 2

be an integer and

f_1(x), f_2(x), \ldots, f_{n}(x)

a sequence of polynomials with integer coefficients. One is allowed to make moves

M_1, M_2, \ldots

as follows: in the

k

-th move

M_k

one chooses an element

f(x)

of the sequence with degree of

f

at least

2

and replaces it with

(f(x) - f(k))/(x-k)

. The process stops when all the elements of the sequence are of degree

1

. If

f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1

, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of

n

identical polynomials of degree 1.

Problem Statement