When deleting either the hundreds or the units digit from a three-digit (integer) number, the remaining two-digit number is 8 times larger than the number written with the deleted digit. What is the ratio between the two-digit number written with the remaining digits and the one-digit number written with the deleted digit, if we delete the middle digit?
I. solution: Let the three-digit number in question be: $\overline{a b c}$, where $a \neq 0$. From the two conditions, we get the system of equations for the three unknown digits:

$$
\begin{aligned}
& (\overline{b c}=) 10 b+c=8 a \\
& (\overline{a b}=) 10 a+b=8 c
\end{aligned}
$$

From here, the digits cannot be determined, but for example, $a$ and $c$ can be expressed in terms of $b$ as a parameter:

$$
\begin{aligned}
& a=\frac{3 b}{2} \\
& c=2 b
\end{aligned}
$$

Since $a \neq 0$, $b$ cannot be 0 either, so the sought ratio is:

$$
\frac{\overline{a c}}{b}=\frac{10 a+c}{b}=\frac{15 b+2 b}{b}=17
$$

Gálfi László (Budapest, Fazekas M. g. I. o. t.)

Most of the solutions first determined the possible numbers. These are the following:

II. solution: According to (1), $b$ can only be even, so according to (2), $c$ is not 0 and a multiple of 4, hence $c=4$ or $c=8$. From this, the appropriate values of $b$ are 2 and 4, and those of $a$ are 3 and 6, so only the numbers 324 and 648 can satisfy the conditions. Indeed, both do. By deleting the middle digit, the same ratio is obtained in both cases: $34: 2=68: 4=17$.

Csákó György (Sátoraljaújhely, Kossuth L. g. I. o. t.)

III. solution: Based on (1) and (2), the number in question is: $100 a+10 b+c=150 b+10 b+2 b=162 b$. Similarly, it can be expressed with $a$ because $b=2 a / 3$, and $c=4 a / 3$:

$$
\overline{a b c}=100 a+10 \cdot \frac{2 a}{3}+\frac{4 a}{3}=108 a
$$

Finally, with $c$, $\overline{a b c}=81 c$. According to these, $\overline{a b c}$ is a multiple of 162 (and thus 81) and 108, so it is also a multiple of their least common multiple, 324. Since it is a three-digit number, only 324, 648, and 972 can come into question. We have already seen the first two, but the third does not meet our requirements.

Gergelics Lajos (Pécs, Zipernovszky K. mechanical engineering t. II. o.t.)

Note. The solution was made possible by the fact that the system of equations with two unknowns is homogeneous, with no term that does not contain an unknown. Thus, the ratio of the unknowns can be calculated, but the values of the digits can only be single-digit (integer) numbers.