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Maximal value of Q(2023) under class of polynomials

Source: 2023 Macedonian Team Selection Test P5

May 21, 2023
algebrapolynomialabstract algebra

Problem Statement

Let Q(x)=a2023x2023+a2022x2022+⋯+a1x+a0∈Z[x]Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x] be a polynomial with integer coefficients. For an odd prime number pp we define the polynomial Qp(x)=a2023p−2x2023+a2022p−2x2022+⋯+a1p−2x+a0p−2.Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}. Assume that there exist infinitely primes pp such that Qp(x)−Q(x)p\frac{Q_{p}(x)-Q(x)}{p} is an integer for all x∈Zx \in \mathbb{Z}. Determine the largest possible value of Q(2023)Q(2023) over all such polynomials QQ.
Proposed by Nikola Velov
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