3. Suppose that each of the two banks $A$ and $B$ will have a constant annual interest rate over the next two years. If we deposit 5/6 of our savings in bank $A$ and the remainder in bank $B$, our savings will grow to 67000 Kč after one year and to 74900 Kč after two years. However, if we deposit 5/6 of our savings in bank $B$ and the remainder in bank $A$, our savings will grow to 71000 Kč after one year. To what amount would our savings increase after two years in the latter case?
SOLUTION. Let our initial savings be $x$ Kč and let the annual interest rate $u$ of bank A (bank B) be $p\% (q\%)$, i.e., a deposit with bank A (bank B) will increase $a$-fold ($b$-fold) after one year, where $a=1+\frac{p}{100}\left(b=1+\frac{q}{100}\right)$. According to the problem, we have

$$
\begin{gathered}
\left(\frac{5}{6} \cdot x\right) \cdot a+\left(\frac{1}{6} \cdot x\right) \cdot b=67000 \\
{\left[\left(\frac{5}{6} \cdot x\right) \cdot a\right] \cdot a+\left[\left(\frac{1}{6} \cdot x\right) \cdot b\right] \cdot b=74900} \\
\left(\frac{1}{6} \cdot x\right) \cdot a+\left(\frac{5}{6} \cdot x\right) \cdot b=71000
\end{gathered}
$$

and after rearrangement,

$$
\begin{gathered}
5 \cdot \frac{x a}{6}+\frac{x b}{6}=67000 \\
5 \cdot \frac{x a}{6} \cdot a+\frac{x b}{6} \cdot b=74900 \\
\frac{x a}{6}+5 \cdot \frac{x b}{6}=71000
\end{gathered}
$$

If we denote $u=\frac{1}{6} x a$ and $v=\frac{1}{6} x b$, the first and third equations in the system become

$$
\begin{aligned}
& 5 u+v=67000 \\
& u+5 v=71000
\end{aligned}
$$

from which we get $u=11000$ and $v=12000$. Since $a=6 u / x$ and $b=6 v / x$, the second equation of the system can be written as

$$
\frac{5}{6} \cdot x \cdot \frac{36 u^{2}}{x^{2}}+\frac{1}{6} \cdot x \cdot \frac{36 v^{2}}{x^{2}}=74900
$$

or

$$
\frac{30 u^{2}+6 v^{2}}{x}=74900
$$

from which, for $u=11000$ and $v=12000$, we get $x=60000$, thus

$$
\begin{aligned}
& a=\frac{6 u}{x}=\frac{66000}{60000}=1.1, \\
& b=\frac{6 v}{x}=\frac{72000}{60000}=1.2 .
\end{aligned}
$$

The sought amount is therefore

$$
\left(\frac{1}{6} \cdot x \cdot a^{2}+\frac{5}{6} \cdot x \cdot b^{2}\right) \mathrm{K} \check{c}=\left(10000 \cdot 1.1^{2}+50000 \cdot 1.2^{2}\right) \mathrm{K} \check{c}=84100 \mathrm{Kč.}
$$

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