HMMT February 2020 #0kbn

HMMT February 2020Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Problem: 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.