HMMT February #0ku4

HMMT FebruaryHard

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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.

Roots of unitySymmetric functionsComplex numbers

Problem

Problem: Let $\zeta = e^{2 \pi i / 99}$ and $\omega = e^{2 \pi i / 101}$ . The polynomial

x^{9999} + a_{9998} x^{9998} + \cdots + a_{1} x + a_{0}

has roots $\zeta^{m} + \omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m < 99$ and $0 \leq n < 101$ . Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$ .

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