Let's determine the base of the number system in which 12551 can be written as 30407.
If $x$ is the sought base, then

$$
\begin{gathered}
3 x^{4}+0 \cdot x^{3}+4 x^{2}+0 \cdot x+7=12551 \\
3 x^{4}-4 x^{2}-12544=0
\end{gathered}
$$

from which

$$
x^{2}= \pm 64
$$

so the real and positive value of $x$ is: 8.

(Ede Grün, Pécs.)

The problem was also solved by: Baján A., Bauer E., Breuer P., Czigler I., Dénes M., Döri V., Engler J., Erdős V., Fried E., Grünwald Gy., Koffler B., Lengyel K., Rosenberg E., Schnabel L., Setyerov P., Sichermann F., Spitzer L., Szilárd V., Szóbel I., Vilcsek A., Viola R., Weinberger Gy.