Question 24
Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{a_1, a_2, \dots, a_k\}$ of positive integers denote $S(A) = P(a_1) + P(a_2) + \dots + P(a_k)$. The positive integers $m, n$ are such that $m^{d+1} \mid n$. Prove that the set $\{1, 2, \dots, n\}$ can be split into $m$ disjoint subsets $A_1, A_2, \dots, A_m$ of equal size such that $S(A_1) = S(A_2) = \dots = S(A_m)$.
(Klurman Oleksiy)
(Klurman Oleksiy)