Problems

In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than \(720^{\circ}\).

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?

Twelve lines are drawn on the plane, passing through a point \(A\). Prove that there are two of them with angle less than \(17^{\circ}\) between them.