2006 Putnam A5

Let $n$ be a positive odd integer and let $\theta$ be a real number such

that $\theta/\pi$ is irrational. Set $a_k = \tan (\theta + k \pi/n)$,

$k=1,2,\dots,n$. Prove that

$$

\frac{a_1 + a_2 + \cdots + a_n}{a_1 a_2 \cdots a_n}

$$

is an integer, and determine its value.

设 $n$ 为正奇整数，$\theta$ 为实数，例如

$\theta/\pi$ 是无理数。设 $a_k = \tan (\theta + k \pi/n)$,

$k=1,2,\点,n$。证明

$$

\frac{a_1 + a_2 + \cdots + a_n}{a_1 a_2 \cdots a_n}

$$

是一个整数，并确定它的值。