Someone left in their will that a certain amount of money should be selected from their estate and placed in a 4.5% interest-bearing account, from which their hometown community would receive an annual dividend of 5000 frt for 26 years. The heirs challenged the will, so it was only after 3 years that they could proceed with its execution. According to the court's judgment, the community was free to choose either to take the accrued and capitalized 3 installments at once, or to extend the 5000 frt annual dividend for a longer period. The community's mayor chose the latter. For how many years will the dividend apply?
The value of the annuity on the day the final order becomes effective:

$$
A=\frac{a}{k^{26}} \frac{k^{26}-1}{k-1}
$$

Its value after 3 years:

$$
B=\frac{a}{k^{23}} \frac{k^{26}-1}{k-1}
$$

From this, the community is entitled to an annuity of 5000 frtn in $n$ installments, paid in annual advance installments, the value of which at the beginning of the 4th year from the date the final order becomes effective is

$$
B=\frac{a}{k^{n-1}} \frac{k^{n}-1}{k-1}
$$

Therefore

$$
\begin{gathered}
\frac{a}{k^{23}} \frac{k^{26}-1}{k-1}=\frac{a}{k^{n-1}} \frac{k^{n}-1}{k-1} \\
\left(k^{26}-1\right) k^{n-1}=\left(k^{n}-1\right) k^{23} \\
k^{n+25}-k^{n-1}=k^{n+23}-k^{23} \\
k^{n+25}-k^{n+23}-k^{n-1}=-k^{23} \\
k^{n}+k^{n+24}-k^{n+26}=k^{24} \\
k^{n}\left(1+k^{24}-k^{26}\right)=k^{24} \\
k^{n-24}=\frac{1}{1+k^{24}-k^{26}} \\
k=1045 \\
\log k=0 \cdot 01912 \\
\log k^{24}=0 \cdot 45888 \\
\log k^{26}=0 \cdot 49712 \\
k^{24}=2 \cdot 8766 \\
k^{26}=3 \cdot 1414 \\
k^{n-24}=\frac{1}{0 \cdot 6352} \\
(n-24) \log k=-\log 0 \cdot 6352 \\
n-24=-\frac{-0 \cdot 80291+1}{0 \cdot 01912} \\
=\frac{0 \cdot 19809}{0 \cdot 01912} \\
=10 \cdot 36 \\
n=3436
\end{gathered}
$$

(Sándor Kugel, chief gymn. VIII. class student, Losoncz).