Solution combinatoric problem -Shortlist Problem no.2 –

International mathematic Olympiad 2013 Problem 2 : In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw k lines not passing through the...

Combinatorics problem in IMO 2013

combinatoric problem in international mathematic olympiad ( IMO) 2013 Problem 1: Let n be a positive integer. Find the smallest integer k with the following property: Given any real numbers a1, . . . , ad such that a1...

The possible partition into some parts

Solution IMO 2015 (combinatoric shortlist ) Problem 7: In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least 3, and...

Proving that there infinitely positive integer that are not clean (unique representation)

Solution IMO 2015 (combinatoric shortlist problem ) Problem 6: Let S be a nonempty set of positive integers. We say that a positive integer n is clean if it has a unique representation as a sum of an odd...

prove that there exist two positive integers from the infinite sequence

solution IMO 2015 (combinatoric shortlist problem) Problem 5: Consider an infinite sequence a1, a2, …. of positive integers with for all . Suppose that for any two distinct indices i and j we have . Prove that there exist...

Combinatoric : problem about determine the outcome of the game

Solution IMO 2015 (shortlist combinatoric problem) Problem 4: Let n be a positive integer. Two players A and B play a game in which they take turns choosing positive integers . The rules of the game are: A player...