Let the radius of the sphere be $r$, the radius of the base of the cone be $x$, the slant height of the cone be $l$, and the height of the illuminated spherical cap be $r-y$. The lateral surface area of the cone is $P=x \pi l$, and the surface area of the illuminated spherical cap is $F=2 r \pi(r-y)$. The values of $x$ and $y$ are determined as follows:

$$
x^{2}=r^{2}-y^{2}=(4 r-y) y
$$

from which

$$
y=\frac{r}{4} \text { and } x=\frac{r}{4} \sqrt{15}
$$

and since $l=4 x$, therefore $l=r \sqrt{15}$. Using these data:

$$
P=\frac{15 r^{2} \pi}{4} \text { and } F=\frac{3 r^{2} \pi}{2}
$$

why is it that:

$$
F: P=2: 5
$$

(István Bella.)

The problem was also solved by: Barna D., Döme B., Freibauer E., Juvancz I., Koós A., Krausz B., Gy. Krisztián, Gy. Lukhaub, R. Obláth, J. Prohászka, G. Sasvári, J. Weisz.
The distance of a light point from a sphere is equal to three times the radius of the sphere. How does the illuminated part of the sphere's surface relate to the lateral surface of the light cone?