Ageev C.M.

A road 1 km long is fully illuminated by street lamps, each of which illuminates a section of the road 1 m long. What is the maximum number of street lamps that can be on the road, given that after turning off any street lamp, the road will no longer be fully illuminated? #
Let's number the street lamps with natural numbers in the order of their placement along the road. If the segments illuminated by the $n$-th and $(n+2)$-th lamps intersect (at least at one point), then the $(n+1)$-th lamp can be turned off.

Therefore, segments with different odd numbers do not intersect. On a segment of length 1000 meters, no more than 999 non-intersecting segments of length 1 meter can be placed. If there were at least 1999 lamps, then there would be at least 1000 lamps with odd numbers. Thus, there are no more than 1998 lamps.

Let's place 1998 lamps such that the centers of the illuminated segments form an arithmetic progression, where the first term is $\frac{1}{2}$ meters, and the 1998-th term is $999 \frac{1}{2}$ meters. (The common difference of this progression is $\frac{999}{1997}$.) The distance between the $n$-th and $(n+2)$-th lamps is $\frac{1998}{1997}$. Therefore, there is a gap of $\frac{1}{1997}$ meters between the segments illuminated by these lamps. This gap is illuminated only by the $(n+1)$-th lamp. Therefore, no lamp can be turned off.