The name of a river consists of three letters. The three numbers, which represent the positions of the letters in the $A B C$ (excluding J), add up to 35. If we increase the third number by one, the first two numbers and the new third number form an arithmetic sequence; if we add 3 to the first number, the new number and the second and third numbers form a geometric sequence.
In the sense of the task:

$$
\begin{gathered}
x+y+z=35 \\
x-y=y-(z+1) \\
\frac{x+3}{y}=\frac{y}{z}
\end{gathered}
$$

From these equations:

$$
x_{1}=15 y_{1}=12, z_{1}=8 \text { and } x_{2}=5 y_{2}=12, z_{2}=18
$$

Since there is no river named Pmh, the sought river is: Ems.

(Benedek Zsolt, Budapest.)

The task was also solved by: Appel S., Barcsay Á., Bayer B., Bender E., Burján K., Csete F.A., Czank K., Engel D., Engel R., Faith F., Fleischer F., Filkorn J., Freibauer E., Jankovich S., Kerekes T., Kiss A., Kohn B., Kornis F., Krausz B., Krisztián Gy., Lindtner M., Lukhaub Gy., Lupsa Gy., Obláth R., Pollák L., Rozlosnik P., Sasvári G., Spitzer V., Stern D., Stromfeld F., Szentiványi L., Szibelth S., Szőllősy J., Téglás G., Weisz J.