Someone borrows 20,000 francs at a $4\%$ interest rate and commits to repaying it in 10 equal annuities, with the first one three years after the loan is taken out. What is the amount of the annuity? The debtor dies after the fifth annuity payment and the heirs prefer to pay off the entire debt in one lump sum two years after the death! How much will they have to pay?

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The borrowed amount is $t$, the annual annuity to be paid over ten years is $r$, and the interest factor is $e$. Therefore, the present values of the individual annuities are:

$$
\frac{r}{e^{3}}, \frac{r}{e^{4}}, \ldots \frac{r}{e^{12}}
$$

so

$$
t=\frac{r}{e^{3}}+\frac{r}{e^{4}},+\ldots+\frac{r}{e^{12}}
$$

or

$$
t=\frac{r}{e^{12}} \cdot \frac{e^{10}-1}{e-1}
$$

from which

$$
r=t e^{12} \cdot \frac{e-1}{e^{10}-1}
$$

The value of the 5 annuities still to be paid after death is, at the time of death:

$$
\frac{r}{e^{5}} \cdot \frac{e^{5}-1}{e-1}
$$

and the value of this amount two years later is:

$$
x=\frac{r}{e^{3}} \cdot \frac{e^{5}-1}{e-1}
$$

Substituting the value of $r$ from (1):

$$
x=t e^{12} \cdot \frac{e-1}{e^{10}-1} \cdot \frac{e^{5}-1}{e^{3}(e-1)}=t \cdot \frac{e^{9}}{e^{5}+1}
$$

Substituting the given values:

$$
r=2668 \text { forints and } x=12842 \text{ forints}.
$$

(Singer Arnold, Losoncz.)

The problem was also solved by: Bayer B., Filkorn J., Krausz B., Krisztián Gy., Póka Gy., Sasvári G., Sasvári J.