19. (26th Russian Mathematical Olympiad, 11th Grade, Problem 1) Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfy the inequality \( f(x+y) + f(y+z) + f(z+x) \geqslant 3 f(x+2y+3z) \) for all \( x, y, z \in \mathbf{R} \).
19. Let $x=y=-z$, we get $f(2 x)+f(0)+f(0) \geqslant 3 f(0)$,

thus $f(2 x) \geqslant f(0)$. On the other hand, let $x=z=-y$, we get $f(0)+f(0)+f(2 x) \geqslant 3 f(2 x)$, thus $f(2 x) \leqslant$ $f(0)$. Therefore, $f(0) \geqslant f(2 x) \geqslant f(0)$. That is, $f(2 x) \equiv$ constant $=c$.
It is easy to verify that for any $c \in \mathbf{R}$, the function $f(x) \equiv c$ satisfies the condition.