Question 3
For every polynomial $f(x)=a_0+a_1x+\cdots+a_mx^m$, define $\Gamma(f)=a_0^2+a_1^2+\cdots+a_m^2$. Let $P(x)=(x+1)(x+2)\cdots(x+2020)$. Prove that there exist at least $2^{2019}$ pairwise distinct polynomials $Q_k(x)$, all with positive coefficients, satisfying $\deg Q_k=2020$ and $\Gamma(Q_k(x)^n)=\Gamma(P(x)^n)$ for every positive integer $n$.