Two points, the centers of two circles, move at a constant speed towards the vertex of a right angle along its legs. The radii of the circles are $9 \mathrm{~m}$ and $4 \mathrm{~m}$; their initial distances from the vertex of the right angle are $48 \mathrm{~m}$ and $14 \mathrm{~m}$, respectively; after $9$ seconds, the circles touch each other externally, and after an additional $2$ seconds, they touch each other internally; at what speed are they moving?
If the speed of one point is $x$, and the other's is $y$

$$
\begin{gathered}
(48-9 x)^{2}+(14-9 y)^{2}=169 \\
(48-11 x)^{2}+(14-11 y)^{2}=25
\end{gathered}
$$

These two equations can also be written as:

$$
\begin{aligned}
81\left(x^{2}+y^{2}\right)-36(24 x+7 y) & =-2331 \\
121\left(x^{2}+y^{2}\right)-44(24 x+7 y) & =-2475
\end{aligned}
$$

From here,

$$
\begin{aligned}
x^{2}+y^{2} & =17 \\
24 x+7 y & =103
\end{aligned}
$$

Solving the system of equations, we get

$$
x_{1}=4 \mathrm{~m}, \quad y_{1}=1 \mathrm{~m}
$$

and

$$
x_{2}=3.9104 \mathrm{~m}, \quad y_{2}=1.3072 \mathrm{~m}
$$

(Odön Kornis.)

The problem was also solved by: F. Bojedain, M. Devecis, E. Freibauer, J. Kárf, E. Probst, M. Roth, Ö. Spitzer, I. Szabó.