Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:

\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]

Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
To solve the problem, we need to find the necessary and sufficient condition for the inequality 

\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0\]

to hold for all real numbers \(x, y, z\). 

1. **Assume \(x \geq y \geq z\):**
   Without loss of generality, we can assume \(x \geq y \geq z\). Let \(a = x - y\) and \(b = y - z\). Note that \(a \geq 0\) and \(b \geq 0\). Then, \(x - z = a + b\).

2. **Rewrite the inequality:**
   Substitute \(a\) and \(b\) into the inequality:
   \[
   A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0
   \]
   becomes
   \[
   A(a)(a+b) + B(b)(-a) + C(-(a+b))(-b) \geq 0
   \]
   Simplify the terms:
   \[
   A(a^2 + ab) - B(ab) + C(b^2 + ab) \geq 0
   \]
   \[
   A a^2 + A ab - B ab + C b^2 + C ab \geq 0
   \]
   Combine like terms:
   \[
   A a^2 + (A - B + C) ab + C b^2 \geq 0
   \]

3. **Analyze the quadratic form:**
   The expression \(A a^2 + (A - B + C) ab + C b^2 \geq 0\) must be non-negative for all \(a \geq 0\) and \(b \geq 0\). This is a quadratic in \(a\) and \(b\). For this quadratic to be non-negative, the discriminant must be non-positive.

4. **Discriminant condition:**
   The discriminant of the quadratic \(A a^2 + (A - B + C) ab + C b^2\) with respect to \(a\) is:
   \[
   \Delta = (A - B + C)^2 - 4AC
   \]
   For the quadratic to be non-negative, we require:
   \[
   \Delta \leq 0
   \]
   Therefore,
   \[
   (A - B + C)^2 - 4AC \leq 0
   \]

5. **Simplify the condition:**
   \[
   (A - B + C)^2 \leq 4AC
   \]
   Taking the square root of both sides (noting that both sides are non-negative):
   \[
   |A - B + C| \leq 2\sqrt{AC}
   \]

Thus, the necessary and sufficient condition for the inequality to hold for all real numbers \(x, y, z\) is:

\[
|A - B + C| \leq 2\sqrt{AC}
\]

The final answer is \( \boxed{ |A - B + C| \leq 2\sqrt{AC} } \).