OIM 1992 Problem 2

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$ , consider the function $$f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}$$ Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$ .