Number of real roots of $\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+...+\frac{a_n}{a_n-x}=2016$ for $0<a_1<...<a_n$?

Let $f(x)$ be the left side. Note that $f(x) \to 0$ as $x \to \pm \infty$, $+\infty$ as $x \to a_i-$ and $-\infty$ as $x \to a_i+$. So there will be at least one root in each interval $(-\infty, a_1)$, $(a_1, a_2)$, ..., $(a_{n-1}, a_n)$. Since the equation is equivalent to a polynomial equation with degree $n$, there are at most $n$ roots. Therefore there are exactly $n$ real roots.

EDIT: There is nothing special about $2016$: any other positive real number would work (for a negative number, you would replace $(-\infty, a_1)$ by $(a_n, \infty)$ and still have $n$ real roots). For $0$, on the other hand, there would only be $n-1$ roots.