$A$ and $B$ provided a total of $1500 \mathrm{~K}$ for a joint business; $A$ provided the money for 3 months, and $B$ provided it for 4 months. How much did each contribute if, after the business was completed, $A$ received $690 \mathrm{~K}$ including profit, and $B$ received $1080 \mathrm{~K}$?
If $A$ gave $x K$, then $B$ gave $1500-x K$, $A$'s profit is $690-x$, and $B$'s profit is $1080-(1500-x)=x-420$.

According to the problem,

$$
690-x: x-420-3 x: 4(1500-x)
$$

or

$$
x^{2}-7500 x+414000=0
$$

from which

$$
x_{1}=6900 ; \quad x_{2}=600
$$

Only $x_{2}$ satisfies the problem, so $A$ gave $600 K$ and $B$ gave $900 K$.

(Pál Sárközy, Pannonhalma.)

The problem was also solved by: Chambrée M., Engler J., Erdős V., Fried E., Friedrich P., Grün E., Koffler B., Kovács Gy., Lázár B., Miklóssy K., Neumann L., Rajz E., Rosenthal M., Szilárd V., Tandlich E., Viola R.