Berkeley Math Circle Take-Home Contest #1 #0hqm

Berkeley Math Circle Take-Home Contest #1Hard

Community difficultyNo ratings yetBe the first to rate this problem.

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.

Vieta's formulasRoots of unityComplex numbers

Problem

Problem:

Let $z = \cos \dfrac{2\pi}{n} + i \sin \dfrac{2\pi}{n}$ where $n$ is a positive odd integer. Prove that

\frac{1}{1+z} + \frac{1}{1+z^{2}} + \frac{1}{1+z^{3}} + \cdots + \frac{1}{1+z^{n}} = \frac{n}{2}

← Back to all problems