Problem:
Four lighthouses are arbitrarily placed in the plane. Each has a stationary lamp which illuminates an angle of degrees. Prove that the lamps can be rotated so that at least one lamp is visible from every point of the plane.
Solution
Solution:
Take a north direction, arbitrary except that no points are aligned north-south or east-west. Take the two most northerly points. Point the lamp for the more easterly of these two in the direction SW (so that it covers directions S to W). Point the lamp for the other in the direction SE. For the other two points, point the lamp for the more easterly in the direction NW, and the lamp for the other in the direction NE.
Clearly the lamps cover all directions, the only possible problem is uncovered strips. However, the two lamps pointing N are below the two lamps pointing S, and the two lamps pointing E are west of the two lamps pointing W, so there are no uncovered strips.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.