Question 43
We define $\mathbb{F}_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{101}$ (the integers modulo $101$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100 x+5)=100 x^{2}+2 x+15$ in $\mathbb{F}_{101}[x]$ because the corresponding coefficients are equal modulo $101$.
We say that $f(x) \in \mathbb{F}_{101}[x]$ is lucky if it has degree at most $1000$ and there exist $g(x), h(x) \in \mathbb{F}_{101}[x]$ such that
$$
f(x)=g(x)\left(x^{1001}-1\right)+h(x)^{101}-h(x)
$$
in $\mathbb{F}_{101}[x]$. Find the number of lucky polynomials.
We say that $f(x) \in \mathbb{F}_{101}[x]$ is lucky if it has degree at most $1000$ and there exist $g(x), h(x) \in \mathbb{F}_{101}[x]$ such that
$$
f(x)=g(x)\left(x^{1001}-1\right)+h(x)^{101}-h(x)
$$
in $\mathbb{F}_{101}[x]$. Find the number of lucky polynomials.