1. Let real numbers $A, B, C$ be such that the following inequality (＊) holds for any real numbers $x, y, z$. What conditions should $A$, $B, C$ satisfy? (The conditions should be necessary and sufficient, and are not limited to being expressed by equations or inequalities involving only $A, B, C$)
$$
\begin{array}{l}
A(x-y)(x-z)+B(y-z)(y-x) \\
+C(z-x)(z-y) \geqslant 0 .
\end{array}
$$
Let $x=y$ in ( * ), we get
$$
C(z-x)^{2} \geqslant 0 \text {. }
$$

By the arbitrariness of $x, z$, we have $C \geqslant 0$. Similarly, by symmetry, we get
$$
A \geqslant 0, B \geqslant 0, C \geqslant 0 .
$$

Let $x-y=s, y-z=t$, then $x-z=s+t$, ( * ) is equivalent to
$$
\begin{array}{l}
A_{s}(s+t)+B_{t}(-s)+C(-s-t)(-t) \\
\geqslant 0,
\end{array}
$$

or $A s^{2}+(A-B+C) s t+C t^{2} \geqslant 0$.
Obviously, ( * ) holds for any $x, y, z$ if and only if (2) holds for any $s, t$. Also, by condition (1), (2) holds for any $s, t$ if and only if
$$
\Delta=(A-B+C)^{2}-4 A C \leqslant 0,
$$

which is $A^{2}+B^{2}+C^{2} \leqslant 2(A B+B C+C A)$.

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