The diameter of a semicircle is $AB$, its center is $O$. We draw semicircles over $AO$ and $BO$. Calculate the radius of the circle that touches the 3 semicircles. What are the sides of the triangle determined by the points of tangency?

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Let $A O=R$, the radius of the tangent circle be $r$, its center $O^{\prime}$, the midpoints of $A O$ and $O B$ be $C$ and $D$, the base of the triangle to be determined be $2 a$, and one of its sides be $b$.

$1^{\circ}$. From $\triangle C O O^{\prime}$:

$$
\left(\frac{R}{2}+r\right)^{2}=(R-r)^{2}+\left(\frac{R}{2}\right)^{2}
$$

from which

$$
r=\frac{R}{3}
$$

$2^{\circ}$. $\triangle O^{\prime} E F \sim \triangle O^{\prime} C D$, so

$$
\frac{R}{3}: 2 a=\frac{5 R}{6}: R
$$

thus

$$
2 a=\frac{2 R}{5}
$$

If the height of the triangle is $m$, then

$$
m=r+x
$$

and

$$
x=\sqrt{\left(\frac{R}{3}\right)^{2}-\left(\frac{R}{5}\right)^{2}}=\frac{4 R}{5}
$$

so

$$
m=\frac{9 R}{15}=\frac{3 R}{5}
$$

and finally

$$
b=\sqrt{a^{2}+m^{2}}=\frac{R}{5} \sqrt{10}
$$

(Ferencz Kertész, Szeged.)

The problem was also solved by: Aczél F., Bartók I., Bayer B. Demjén E., Deutsch I., Dózsa Zs., Enyedi B., Grób I., Holzmann I., Izsáky L., Kamenitzky M., König D., Messik G., Moskovits Zs., Pilczer P., Póka Gy., Radó I., Riesz K., Riesz M., Schmidl I., Simon S., Szávay Z., Steiner D., Steiner M., Tóbiás L., Vaska I., Wohlstein S.