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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

Problem Statement