15. (26th Russian Mathematical Olympiad, 11th Grade Problem) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, which for all $x, y, z \in \mathbf{R}$ satisfy the inequality $f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+3 z)$.
15. 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, $f(0) \geqslant f(2 x) \geqslant f(0)$, i.e., $f(2 x)=$ constant $=c$.

It is easy to verify that for any $c \in \mathbf{R}$, the function $f(x) \equiv c$ satisfies the condition.