Putnam 2006 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.