Let's start by understanding the problem statement correctly. We have a sequence defined by
$$S_r = x^r + y^r + z^r$$
where $x, y,$ and $z$ are real numbers. We are informed that if $S_1 = x + y + z = 0$, then the following relationship holds:
$$(*)\quad \frac{S_{m+n}}{m+n} = \frac{S_m}{m} \cdot \frac{S_n}{n}$$
for specific pairs $(m,n)$ which are $(2,3), (3,2), (2,5),$ and $(5,2)$.

We aim to determine any other pairs $(m,n)$ for which $(*)$ holds for all real numbers $x, y, z$ with the condition $x + y + z = 0$.

### Analysis

Given $S_1 = x + y + z = 0$, we derive that for any powers $r$ we have:
$$S_1 = 0$$

This condition implies symmetries in the polynomials involved, since the sum of the variables $x, y,$ and $z$ is zero.

From the given $(*)$ relationship, we need to satisfy:
$$\frac{x^{m+n} + y^{m+n} + z^{m+n}}{m+n} = \left(\frac{x^m + y^m + z^m}{m}\right) \left(\frac{x^n + y^n + z^n}{n}\right).$$

This can be rephrased in terms of sums of powers of roots, which hint towards symmetric polynomials and potential applications of elementary symmetric polynomials.

### Verification of Known Pairs

For the pairs $(2, 3)$, $(3, 2)$, $(2, 5)$, and $(5, 2)$:
- $(m, n) = (2, 3)$ and $(3, 2)$ leverage symmetry and repeat similar steps due to their interchangeability.
- Similarly, $(m, n) = (2, 5)$ and $(5, 2)$ are handled analogously, ensuring the expression's symmetry.

Given the constraints and the structural dependencies of powers when $x + y + z = 0$, these form self-consistent symmetric polynomial structures only satisfying the original four pairs.

### Conclusion

After exploring the stated known pairs, testing similar logic for additional pairs did not lead to any additional solutions. The relationship $(*)$ seems to hold uniquely for the symmetric consideration in these specific cases.

Therefore, the pairs for which the equation $(*)$ holds for all real numbers $x, y, z$ satisfying $x + y + z = 0$ remain as:
$$\boxed{(2, 3), (3, 2), (2, 5), (5, 2)}$$
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