4. Let and be on segment of an acute triangle such that and . Let and be the points on and , respectively, such that is the midpoint of and is the midpoint of . Prove that the intersection of and is on the circumference of triangle .
5. For every positive integer , Cape Town Bank issues some coins that has value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than . Prove that we can divide the collection into at most groups such that sum of all coins’ value does not exceed .
6. A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large , in any set of lines in general position it is possible to colour at least lines blue in such a way that none of its finite regions has a completely blue boundary.
Note: Results with replaced by will be awarded points depending on the value of the constant