Assume that $x_1, \dots, x_n$ are non-negative real numbers such that $ x_1 + \dots + x_n \in \mathbb Q~~~~~~~~~~~~~~ \text{ and } ~~~~~~~~~~~~~~~x_1 + 2x_2 + \dots + nx_n\in \mathbb Q. $
Does this imply that the numbers $x_1,\dots, x_n$ are rational too?