Number of real solutions. 
Question : Let $\{a_i\}$ be a sequence of real numbers such that $0<a_1<a_2\cdots <a_n$. Show that the equation : 
  $$\frac{a_1}{a_1−x}+\cdots+\frac{a_n}{a_n−x}=2015$$
  has exactly $n$ real solutions.

My try:
I know that this is an nth degree polynomial. But I really have no idea how to show the required.

Hint: the LHS is continuous in each $(a_i,a_{i+1})$ interval. Note that for a small enough $\varepsilon$, $\dfrac{a_i}{a_i-x} \ll 0$ for an $x\in (a_i,a_i+\varepsilon)$ and $\dfrac{a_{i+1}}{a_{i+1}-x} \gg 0$ for an $x\in (a_{i+1}-\varepsilon,a_{i+1})$.
Hint2: The fact that the RHS is nonzero and the fact that the LHS gets close to $0$ for $x \ll a_1$ or $x\gg a_n$ should yield the last real solution.
