HMMT November 2014 #0jis

HMMT November 2014Hard

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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.

Complex numbersPolynomial interpolation: Newton, Lagrange

Problem

Problem:

Let $z$ be a complex number and $k$ a positive integer such that $z^{k}$ is a positive real number other than $1$ . Let $f(n)$ denote the real part of the complex number $z^{n}$ . Assume the parabola $p(n) = a n^{2} + b n + c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$ . Assuming the smallest possible value of $k$ , find the largest possible value of $a$ .

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