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?
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.