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VMO 2022 problem 5 day 2

Source: Vietnam Mathematical Olympiad 2022 problem 5 day 2

March 5, 2022
vmoalgebrapolynomial

Problem Statement

Consider 2 non-constant polynomials P(x),Q(x)P(x),Q(x), with nonnegative coefficients. The coefficients of P(x)P(x) is not larger than 20212021 and Q(x)Q(x) has at least one coefficient larger than 20212021. Assume that P(2022)=Q(2022)P(2022)=Q(2022) and P(x),Q(x)P(x),Q(x) has a root pq≠0(p,q∈Z,(p,q)=1)\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1). Prove that ∣p∣+n∣q∣≤Q(n)−P(n)|p|+n|q|\le Q(n)-P(n) for all n=1,2,...,2021n=1,2,...,2021
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