## 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...