Let's determine the values of $A$ and $B$, non-zero digits, and the value of the exponent $n$ (a natural number) such that in the decimal form of $1 / \overline{AB}^{n}$, the first two significant digits are $A$ and $B$, and they appear after the decimal point and five 0 digits.
We are looking for a two-digit $\overline{A B}$ (integer) number, for which, for a suitable $n$,

$$
\frac{1}{\overline{A B^{n}}}=0.00000 A B \ldots
$$

or

$$
\frac{10^{7}}{\overline{A B^{n}}}=\overline{A B}, \ldots
$$

The number on the left side is thus between $\overline{A B}$ and $\overline{A B}+1$. Let's denote the $\overline{A B}$ number by $x$, then

$$
x \leq \frac{10^{7}}{x^{n}}1,7481$, $\lg 57>1,7558$ so $3 \lg 56+\lg 57>7,0001$.) In the third case, it does not hold, so our problem can only have two solutions:

$$
\begin{array}{lll}
n_{1}=3, & A_{1}=5, & B_{1}=6 ; \\
n_{2}=4, & A_{2}=2, & B_{2}=5 .
\end{array}
$$

Indeed,

$$
\begin{gathered}
\frac{1}{56^{3}}=0.00000569 \ldots \quad \text { and } \\
\frac{1}{25^{4}}=0.04^{4}=0.2^{8}=0.00000256
\end{gathered}
$$

Imre Major (Budapest, I. István Gymn., II. o. t. ) Béla Tarsó (Veszprém, Lovassy L. Gymn., II. o. t. )