MathDB

Determine all positive integers

n

satisfying the following condition: for every monic polynomial

P

of degree at most

n

with integer coefficients, there exists a positive integer

k\le n

and

k+1

distinct integers

x_1,x_2,\cdots ,x_{k+1}

such that

P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})

.
Note. A polynomial is monic if the coefficient of the highest power is one.

Problem Statement