# 2014 IMO Problems

## Problem 1

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

## Problem 2

Let be an integer. Consider an chessboard consisting of unit squares. A configuration of rooks on this board is 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 squares.

## 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 and

Prove that line is tangent to the circumcircle of

## Problem 4

Points and lie on side of acute-angled so that and . Points and lie on lines and , respectively, such that is the midpoint of , and is the midpoint of . Prove that lines and intersect on the circumcircle of .

## Problem 5

For each positive integer , the Bank of Cape Town issues coins of denomination . Given a finite collection of such coins (of not necessarily different denominations) with total value at most , prove that it is possible to split this collection into or fewer groups, such that each group has total value at most .

## Problem 6

A set of lines in the plane is in 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 are; we call these its . Prove that for all sufficiently large , in any set of lines in general position it is possible to colour at least of the lines blue in such a way that none of its finite regions has a completely blue boundary.