Determine the digits $a, b, c, d, e$ such that the two five-digit numbers written with them satisfy the equation $\overline{a b c d e} \cdot 9=$ $\overline{e d c b a}$.
The five-digit number $\overline{a b c d e}$, when multiplied by 9, results in the five-digit number $\overline{e d c b a}$, which is only possible if the value of $a$ is 1.

That is, $\overline{1 b c d e} \cdot 9 = \overline{e d c b 1}$, from which it follows that $e=9$, because the only (single-digit) number that, when multiplied by 9, ends in 1 is 9.

Our equation can be written as:

$$
9 \cdot 10^{4} + 9 \cdot 10^{3} \cdot b + 9 \cdot 10^{2} \cdot c + 9 \cdot 10 \cdot d + 81 = 9 \cdot 10^{4} + 10^{3} \cdot d + 10^{2} \cdot c + 10 \cdot b + 1
$$

After rearranging:

$$
899 b + 80 c + 8 = 91 d
$$

Since $d$ is at least 1 and at most 9, we have:

$$
91 \leq 899 b + 80 c + 8 \leq 819
$$

From this, it follows that $b$ can only be 0, and from (1):

$$
80 c + 8 = 8(10 c + 1) = 91 d
$$

The left side is divisible by 8, so the right side must also be, and since 91 is odd, $d$ must be divisible by 8, and since $d \leq 9$, $d=8$. Then $80 c + 8 = 728$, and thus $c=9$.

The five-digit number is therefore: 10989. Indeed, $10989 \cdot 9 = 98901$; this is the only solution to the problem. Pajna Gabriella (Debrecen, Tóth Á. Gymnasium, 11th grade)

Note. From the initial conditions, it follows that the sought number is divisible by 9, since the product is divisible by 9, and the sum of its digits is equal to the sum of the digits of the original number. If we already know that $\overline{1 b c d 9} \cdot 9 = \overline{9 d c b 1}$, then from this, $b$ can only be 0 or 1, otherwise there would be a remainder in the 4th place value. If $b=1$, then $d=7$, because only $7 \cdot 9 + 8$ results in 1 in the units place. Then $a + b + c + d + e = 1 + 1 + c + 8 + 9 = 19 + c$, which is only divisible by 9 if $c=8$, but $11889 \cdot 9 \neq 98811$. If, however, $b=0$, then $d=8$ must be, because only $8 \cdot 9 + 8$ ends in 0. $1 + 0 + c + 8 + 9 = 18 + c$ means $c=9$. Indeed, $10989 \cdot 9 = 98901$.