There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point. a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least $\lfloor \frac{(a-1)(a-2)}{10} \rfloor$ intersection points. b) Find all $a$ for which the evaluation in a) is the best possible.