Someone receives a surplus at the end of every year for 15 years, and in the first five years, it is $1000 \mathrm{~K}-\mathrm{t}$ annually, in the next five years, it is $1200 \mathrm{~K}-\mathrm{t}$ annually, and in the last five years, it is $1400 \mathrm{~K}-\mathrm{t}$ annually. If they received $1400 \mathrm{~K}-\mathrm{t}$ annually in the first five years and $1200 \mathrm{~K}-\mathrm{t}$ annually in the second five years, how much would the annual surplus be in the last five years? $(p=4$.
Ha az összes járadékokat a jelen időre vonatkoztatjuk, akkor a feladat értelmében:

$$
\begin{aligned}
& \frac{1000}{e^{5}} \cdot \frac{e^{5}-1}{e-1}+\frac{1200}{e^{10}} \cdot \frac{e^{5}-1}{e-1}+\frac{1400}{e^{15}} \cdot \frac{e^{5}-1}{e-1}= \\
& =\frac{1400}{e^{5}} \cdot \frac{e^{5}-1}{e-1}+\frac{1200}{e^{10}} \cdot \frac{e^{5}-1}{e-1}+\frac{x}{e^{15}} \cdot \frac{e^{5}-1}{e-1}
\end{aligned}
$$

Miből

$$
x=1400-400 e^{10}=807.95 \mathrm{~K}
$$

(Kürth Richárd, Nyitra.)

---

If we consider all the annuities to the present time, then according to the problem:

$$
\begin{aligned}
& \frac{1000}{e^{5}} \cdot \frac{e^{5}-1}{e-1}+\frac{1200}{e^{10}} \cdot \frac{e^{5}-1}{e-1}+\frac{1400}{e^{15}} \cdot \frac{e^{5}-1}{e-1}= \\
& =\frac{1400}{e^{5}} \cdot \frac{e^{5}-1}{e-1}+\frac{1200}{e^{10}} \cdot \frac{e^{5}-1}{e-1}+\frac{x}{e^{15}} \cdot \frac{e^{5}-1}{e-1}
\end{aligned}
$$

From which

$$
x=1400-400 e^{10}=807.95 \mathrm{~K}
$$

(Richárd Kürth, Nyitra.)