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Z->Z polynomial, summation with +-1 coefficients

Source: IMOC 2018 N6

August 18, 2021
algebrapolynomialnumber theory

Problem Statement

If ff is a polynomial sends Z\mathbb Z to Z\mathbb Z and for n∈N≥2n\in\mathbb N_{\ge2}, there exists x∈Zx\in\mathbb Z so that n∤f(x)n\nmid f(x), show that for every k∈Zk\in\mathbb Z, there is a non-negative integer tt and a1,…,at∈{−1,1}a_1,\ldots,a_t\in\{-1,1\} such that a1f(1)+…+atf(t)=k.a_1f(1)+\ldots+a_tf(t)=k.
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