1. Given $\left|x_{i}\right|<1(i=1,2, \cdots, n)$, let $\sum_{i=1}^{n}\left|x_{1}\right|=19+\left|\sum_{k=1}^{n} x_{k}\right|$. Find the minimum value of $n$.
(Given: $\left|\sum_{i=1}^{n} x_{i}\right| \geqslant 0$, then $\sum_{i=1}^{n}\left|x_{i}\right| \geqslant 19$, since $\left|x_{i}\right|<1$, then $n \geqslant 20$. Taking $x_{i}=\frac{19}{20}(i=1,2, \cdots, 10), x_{j}=-\frac{19}{20}(j=11,12, \cdots, 20)$, we have $n=20$, hence $n_{\min }=20$.)