AoPS - Problem

Source: VMO 2019

Tags: polynomial

yumeidesu

16.04.2019 00:10

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 $n$.

*I think it should be: Show that there exists at least $2^{2019}$ polynomials $Q_k(x)$ ($1\le k\le 2^{2019}$) which are pairwise distinct and ... This might help: If $f,g$ are polynomials such that $$f(x)f(1/x)=g(x)g(1/x)$$then $$\Gamma(f(x)^n)=\Gamma(g(x)^n)$$holds for every positive integer $n$. I think this works: Pick any subset $A$ of $S=\{1,2,3,\cdots, 2019\}.$ Say $\{i_1, i_2, \cdots, i_k\}.$ Now change $(x+i_1)(x+i_2)\cdots(x+i_k)(x+2020)$ to $(1+i_1 x)(1+i_2 x)\cdots(1+i_k x)(1+2020x).$ Keep other factors same.