FI-MO 2008 Problem 5

The closed line segment $I$ is covered by finitely many closed line segments. Show that one can choose a subfamily $S$ of the family of line segments having the properties: (1) the chosen line segments are disjoint, (2) the sum of the lengths of the line segments of S is more than half of the length of $I.$ Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".