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)
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)
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)
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)
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)
Let be an infinite sequence of positive integers. Prove that there exists a unique integer such that
IMO 2014 problem 1 (Day 1)
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).