## IMO 2014 Problem 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 .

**IMO 2014 Problem 6 (Day 2)**

## IMO 2014 Problem 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 100 groups such that sum of all coins’ value does not exceed 1.

**IMO 2014 Problem 5 (Day 2)**

## IMO 2014 Problem 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 .

**IMO 2014 Problem 4 (Day 2)**

## IMO 2014 Problem 3

Convex quadrilateral has . Point is the foot of the perpendicular from to . Points and lie on sides and , respectively, such that lies inside triangle and

Prove that the line is tangent to the circumcircle of triangle .

**IMO 2014 Problem 3 (Day 1)**

## IMO 2014 Problem 2

Let be an integer. Consider a chessboard consisting of unit squares. A configuration of rooks on this board is *peaceful* if every row and every column contains exactly one rook. Find the greatest positive integer such that for each peaceful configuration of rooks, there is a square which does not contain a rook on any of its unit squares.

**IMO 2014 Problem 2 (Day 1)**

## IMO 2014 Problem 1

Let be an infinite sequence of positive integers. Prove that there exists a unique integer such that

**IMO 2014 problem 1 (Day 1)**

## Existence of Sylow subgroups

Let be a finite group such that where is a prime number, and . Then there exists a subgroup such that . (such a subgroup is called a Sylow subgroup).