Two straight lines intersect at a right angle, and the centers of two circles move toward the intersection point along each line. The radius of the first circle is $981 \mathrm{~m}$, and the speed of its center is $7 \mathrm{~m}$ per second. The distance of the center from the intersection point at the start is $2442 \mathrm{~m}$. The radius of the second circle is $980 \mathrm{~m}$; the speed of its center is $5 \mathrm{~m}$ per second; the center is $1591 \mathrm{~m}$ away from the intersection point at the moment of simultaneous start. How many seconds after the start will the circumferences of the moving circles first touch externally, first before the intersection point, and then beyond the intersection point, assuming that they do not interfere with each other's movement?
Circles touch when the distance between their centers equals the sum of their radii; if the touch occurs $x \, \text{min}$ later, then according to the problem:

$$
(2442-7 x)^{2}+(1591-5 x)^{2}=(981+980)^{2}
$$

or

$$
x^{2}-677 x+62826=0
$$

from which

$$
x_{1}=566, \quad x_{2}=111
$$

Thus, the circles first touch after $111 \, \text{min}$ and secondly after $566 \, \text{min}$.

(Czank Károly, Déva.)

The problem was also solved by: Appel S., Csete F. A., Filkorn J., Freibauer E., Kerekes T., Krausz B., Krisztián Gy., Lukhaub Gy., Neumann J., Sasvári G., Sasvári J., Spitzer Ö., Stromfeld F., Weisz J.