Given a horizontal strip on the plane (its sides are parallel lines) and n lines intersecting the strip. Every two of them intersect inside the strip, and not a triple has a common point. Consider all the paths along the segments of those lines, starting on the lower side of the strip and ending on the upper side with the properties: moving along such a path we are constantly rising up, and, having reached the intersection, we are obliged to turn to another line. Prove that: a) there are not less than n/2 such a paths without common points; b) there is a path consisting of not less than of n segments; c) there is a path that goes along not more than along n/2+1 lines; d) there is a path that goes along all the n lines. combinatoricscombinatorial geometry