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integer polynomial, S(A)=P (a_1)+...+ P (a_k ), S (A_1) = ... = S (A_k )

Source: Ukraine TST 2012 p11

May 1, 2020
polynomialSubsetscombinatorics

Problem Statement

Let PP be a polynomial with integer coefficients of degree dd. For the set A={a1,a2,...,ak}A = \{ a_1, a_2, ..., a_k\} of positive integers we denote S(A)=P(a1)+P(a2)+...+P(ak)S (A) = P (a_1) + P (a_2) + ... + P (a_k ). The natural numbers m,nm, n are such that md+1∣nm ^{d+ 1} | n. Prove that the set {1,2,...,n}\{1, 2, ..., n\} can be subdivided into mm disjoint subsets A1,A2,...,AmA_1, A_2, ..., A_m with the same number of elements such that S(A1)=S(A2)=...=S(Am)S (A_1) = S(A_2) = ... = S (A_m ).
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