We have a three-digit number, where the digit in the units place is the geometric mean of the other digits. The digit in the tens place is $\frac{1}{6}$ of the sum of the other digits. If we subtract 396 from this number, the digits in the result appear in reverse order. Which number is this?
If $100 x+10 y+z$ is the sought number, then

$$
\begin{gathered}
z^{2}=x y \\
y=\frac{x+z}{6}
\end{gathered}
$$

and

$$
100 x+10 y+z-396=100 z+10 y+x
$$

or

$$
x-z=4
$$

The positive integer solution of the equations (1), (2), and (3) is: $x=8, y=2$ and $z=4$, so 824 is the sought number.

(Zoltán Grossberger, Losoncz.)

Number of solutions: 28.