The angle $x O y$ is a right angle. A circle $C$, with radius $R$, touches the $O x$ at $A$; another circle $C^{\prime}$, with radius $R^{\prime}$, touches the $O y$ at $B$ and, in addition, the two circles also touch each other. Furthermore, $O A + O B = m$. Calculate $O A$ and $O B$; construct the two circles.
Let $O A=x$ and $O B=y$. According to the assumption,

$$
x+y=m
$$

However, if we consider the triangle $C D C^{\prime}$, where $D$ is the intersection point of the perpendiculars dropped from $C$ to $O y$ and from $C$ to $O x$, we see that:

$$
\left(x-R^{\prime}\right)^{2}+(y-R)^{2}=\left(R+R^{\prime}\right)^{2}
$$

From 1),

$$
y=m-x
$$

Substituting this into 2), we get:

$$
\left(x-R^{\prime}\right)^{2}+(m-x-R)^{2}=\left(R+R^{\prime}\right)^{2}
$$

or, rearranging,

$$
2 x^{2}-2\left(m+R^{\prime}-R\right) x+m^{2}-2 m R-2 R R^{\prime}=0
$$

from which

$$
x=\frac{1}{2}\left\{m+R^{\prime}-R \pm \sqrt{\left(m+R+R^{\prime}\right)^{2}-2 m^{2}}\right\}
$$

From the equation $y=m-x$, we get:

$$
y=\frac{1}{2}\left\{m+R-R^{\prime} \mp \sqrt{\left(m+R+R^{\prime}\right)^{2}-2 m^{2}}\right\}
$$

The construction of the two circles proceeds as follows: From the right-angled triangle $C D C^{\prime}$, the hypotenuse $C C^{\prime}=R+R^{\prime}$ is known, and the sum of the two legs, $C D+D C^{\prime}=m-R-R^{\prime}$. If I add the side $C D$ to the side $D C^{\prime}$, I obtain a point $C_{1}$ which, together with points $C$ and $C^{\prime}$, defines the triangle $C C_{1} C^{\prime}$. From this, two sides, $C C^{\prime}$ and $C C_{1}$, and the angle $C C_{1} C^{\prime}=45^{\circ}$ are known. The construction yields $0, 1, 2$ solutions, depending on whether $C C^{\prime} \lesseqgtr C I$, where the latter is the perpendicular dropped from $C$ to the ray $C_{1} L$ at an angle of $45^{\circ}$. The perpendiculars dropped from points $C^{\prime}$ and $C_{1}^{\prime}$ to $C_{1} C$ provide the points $D$ and $D_{1}$. The coordinates of these points in the $x O y$ system are $\xi=R^{\prime}, \eta=R$.

(Friedmann Berhát, Gymnasium VII. o. t. S. A. Ujhely)

The problem was also solved by: Grünhut Béla, Pécs; Jankovich György, Fülek; Meitner Elemér, Budapest; Weisz Lipót, Pécs.