Adva van két egymásra merőleges egyenes $O A$ és $O B$; továbbá egy kör, melynek középpontja $C$, sugara $R$ és mely ez adott egyeneseket érinti? Mekkora ama kör sugara, mely az adott egyeneseket érinti és az adott kört derékszög alatt metszi?

Given are two perpendicular lines $O A$ and $O B$; and a circle whose center is $C$, radius is $R$, and which touches these given lines. What is the radius of the circle that touches the given lines and intersects the given circle at a right angle?
Let the center of the sought circle be $C_{1}$, its radius $x$, and one of the intersection points of the two circles be $M$. The two circles intersect at a right angle if $\angle C M C_{1}=90^{\circ}$; in this case,

$$
x^{2}+R^{2}=\overline{C C_{1}^{2}}
$$

But $C C_{1}=C_{1} O-C O=x \sqrt{2}-R \sqrt{2}=(x-R) \sqrt{2}$, so

$$
x^{2}+R^{2}=2(x-R)^{2}
$$

or

$$
x^{2}-4 x R+R^{2}=0
$$

from which

$$
x=R(2 \pm \sqrt{3})
$$

(Henrik Fodor, Beregszász.)

The problem was also solved by: L. Bánó, I. Csada, I. Dömémy, A. Haar, E. Kirchknopf, J. Kiss, P. Merse, O. Morvai, A. Paunz, J. Rosenberg, M. Rosenthal, P. Sárközy, J. Székely, E. Tandlich, the V. district math circle.