For $n=2$ this is obviously possible. For $n>2$, inscribe in the arena a regular $k$-gon, where $k=n(n-1) / 2$. Then we can establish 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 in which it is involved. It is easy to verify that this illumination has the required properties.

## Answer

For all $n \geq 2$.
[ $\left.\frac{\text { Pairings and Groups; Bijections }}{\text { Regular Polygons }}\right]$
[ Examples and Counterexamples. Constructions ]

A circus arena is illuminated by $n$ different spotlights. Each spotlight illuminates a certain convex shape. It is known that if one arbitrary spotlight is turned off, the arena will still be fully illuminated, but if any two arbitrary spotlights are turned off, the arena will no longer be fully illuminated. For which values of $n$ is this possible?