RU-ALLRUSSIAN 1975 Problem 11

S is a horizontal strip in the plane. n lines are drawn so that no three are collinear and every pair intersects within the strip. A path starts at the bottom border of the strip and consists of a sequence of segments from the n lines. The path must change line at each intersection and must always move upwards. Show that: (1) there are at least n/2 disjoint paths; (2) there is a path of at least n segments; (3) there is a path involving not more than n/2 + 1 of the lines; and (4) there is a path that involves segments from all n lines.