Problem

Source: VMO 2019

Tags: polynomial



For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let $$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions: i) $\deg {{Q}_{k}}(x)=2020.$ ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.