19. Given positive real numbers $a$, $b$, $c$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a+b^{2}+2 a c=29, \\
b+c^{2}+2 a b=18, \\
c+a^{2}+2 b c=25 .
\end{array}\right.
$$

Then the value of $a+b+c$ is ( )
(A)9
(B) 10
(C) 11
(D) 7
(E) 8
19.E.

Let $x=a+b+c$. By adding the three equations, we get $x+x^{2}=72$, which simplifies to $(x-8)(x+9)=0$. Since $a, b, c$ are all positive numbers, then $x>0$, hence $a+b+c=8$.
Note that $a=4, b=1, c=3$ satisfies this system of equations.