AT-OEMO-FED1 2010 Problem 2

For a positive integer $n$ , we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$ . For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$ .

(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)