$[$ Convex and non-convex figures (miscellaneous).]

The circus arena is illuminated by $n$ different spotlights. Each spotlight illuminates a convex figure. It is known that if any one spotlight is turned off, the arena will still be fully illuminated, but if any two spotlights are turned off, the arena will not be fully illuminated. For which $n$ is this possible?
For each pair of spotlights in the arena, there must be an area illuminated precisely by these two spotlights.

## Solution

Inscribe a regular $k$-gon in the arena, where $k=1 / 2 n(n-1)$ is the number of different pairs that can be formed from $n$ spotlights. Then, there can be a one-to-one correspondence between the segments cut off by the sides of the $k$-gon and the pairs of spotlights. Let each spotlight illuminate the entire $k$-gon and the segments corresponding to the pairs of spotlights in which it is included. It is easy to verify that this illumination has the required properties.

## Answer

For any $n \geq 2$.