Question 168
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 $x_1, x_2, \dots, x_{k+1}$ such that $P(x_1) + P(x_2) + \dots + P(x_k) = P(x_{k+1})$.
*Note.* A polynomial is *monic* if the coefficient of the highest power is one.
*Note.* A polynomial is *monic* if the coefficient of the highest power is one.