19. Solve the system of equations:

$$
\left\{\begin{array}{l}
x_{0}=x_{0}^{2}+x_{1}^{2}+\ldots+x_{100}^{2} ; \\
x_{1}=2\left(x_{0} x_{1}+x_{1} x_{2}+\ldots+x_{99} x_{100}\right) ; \\
x_{2}=2\left(x_{0} x_{2}+x_{1} x_{3}+\ldots+x_{98} x_{100}\right) \\
x_{3}=2\left(x_{0} x_{3}+x_{1} x_{4}+\ldots+x_{97} x_{100}\right) ; \\
\cdots \cdots \cdots \\
x_{100}=2 x_{1} x_{100}
\end{array}\right.
$$
71.19. Answer. There are a total of 202 solutions. The main series: $x_{0}=0, x_{k}= \pm 1 / 2, x_{j}=0$ for $j \neq k$ (200 solutions). Another solution: all $x_{i}$ are equal to zero, and the last solution is: $x_{0}=$ $=1, x_{i}=0$ for $i>0$.