MathDB

3

Part of 2009 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2009 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Is it possible in a plane mark 1010 red, 1010 blue and 1010 green points (all distinct) such that three conditions hold: i)i) For every red point AA there exists a blue point closer to point AA than any other green point ii)ii) For every blue point BB there exists a green point closer to point BB than any other red point iii)iii) For every green point CC there exists a red point closer to point CC than any other blue point
combinatoricsColoring
Regional Olympiad - FBH 2009 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Decomposition of number nn is showing nn as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer nn show that number of decompositions is 2n−12^{n-1}
Decompositioncombinatorics
Regional Olympiad - FBH 2009 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
There are nn positive integers on the board. We can add only positive integers c=a+ba−bc=\frac{a+b}{a-b}, where aa and bb are numbers already writted on the board. a)a) Find minimal value of nn, such that with adding numbers with described method, we can get any positive integer number written on the board b)b) For such nn, find numbers written on the board at the beginning
boardMinimalcombinatorics