MathDB

5

Part of 2018 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2018 Grade 9 Problem 5

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Let HH be an orhocenter of an acute triangle ABCABC and MM midpoint of side BCBC. If DD and EE are foots of perpendicular of HH on internal and external angle bisector of angle ∠BAC\angle BAC, prove that MM, DD and EE are collinear
geometryangle bisectororthocentercollinear
Regional Olympiad - FBH 2018 Grade 10 Problem 5

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Board with dimesions 2018×20182018 \times 2018 is divided in unit cells 1×11 \times 1. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If WW is number of remaining white chips, and BB number of remaining black chips on board and A=min{W,B}A=min\{W,B\}, determine maximum of AA
boardChipsmaximumcombinatorics
Regional Olympiad - FBH 2018 Grade 11 Problem 5

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
It is given 20182018 points in plane. Prove that it is possible to cover them with circles such that: i)i) sum of lengths of all diameters of all circles is not greater than 20182018 ii)ii) distance between any two circles is greater than 11
Planecombinatorial geometrycombinatoricscovering