9. Let real numbers $A, B, C$ satisfy the inequality $A(x-y)(x-z)+B(y-z)(y-x)+C(z-x)(z-y) \geqslant 0$ (*) for any real numbers $x, y, z$. What conditions should $A, B, C$ meet? (The conditions should be both necessary and sufficient, and should be expressed using only equations or inequalities involving $A, B, C$).
9. Let $x=y$ in the original inequality, we get $C(z-x) \geqslant 0$. By the arbitrariness of $x, y$, we have $C \geqslant 0$. By symmetry, we also have $A \geqslant 0, B \geqslant 0, C \geqslant 0$.
Let $x-y=s, y-z=t$, then $x-z=s+t$. Thus, inequality (*) is equivalent to
$$
A s^{2}+(A-B+C) s t+C t^{2} \geqslant 0 .
$$

Obviously, (*) holds for any $x, y, z$ if and only if the above inequality holds for any $s, t$. According to conditions (1) and (2), the necessary and sufficient condition for the above inequality to hold for any $s, t$ is
$$
A^{2}+B^{2}+C^{2} \leqslant 2(A B+B C+C A) .
$$

In summary, the conditions that $A, B, C$ should satisfy are (1) and (3).