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Putnam 2006 A5

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December 4, 2006
PutnamtrigonometryalgebrapolynomialratioinductionVieta

Problem Statement

Let nn be a positive odd integer and let θ\theta be a real number such that θ/π\theta/\pi is irrational. Set ak=tan⁡(θ+kπ/n), k=1,2…,n.a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n. Prove that a1+a2+⋯+ana1a2⋯an\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}} is an integer, and determine its value.
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