Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.\]What is the smallest possible value of $n$?
The left-hand side satisfies \[|x_1| + |x_2| + \dots + |x_n| < 1 + 1 + \dots + 1 = n,\]while the right-hand side satisfies \[19 + |x_1 + x_2 + \dots + x_n| \ge 19.\]Therefore, $n > 19,$ so $n \ge 20.$ It is possible that $n=20,$ since, for example, we can choose \[\begin{aligned} x_1 = x_2 = \dots = x_{10} &= \tfrac{19}{20}, \\ x_{11} =x_{12} = \dots =x_{20}& = -\tfrac{19}{20}, \end{aligned}\]which makes $|x_1| + |x_2| + \dots = |x_{20}| = 19$ and $|x_1 + x_2 + \dots + x_{20}| = 0.$ Therefore the answer is $\boxed{20}.$