3. 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?
3. 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 street lamps intersect (at least at one point), then the $(n+1)$-th street lamp can be turned off. Therefore, segments with different odd numbers do not intersect. On a segment of length 1000 m, it is impossible to place more than 999 non-intersecting segments of length 1 m. If there were at least 1999 street lamps, then there would be at least 1000 street lamps with odd numbers. Thus, there are no more than 1998 street lamps.

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

Comment. Compare with problem 6 for 8th grade.