MathDB

4

Part of 2010 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2010 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
It is given set with n2n^2 elements (n≥2)(n \geq 2) and family F\mathbb{F} of subsets of set AA, such that every one of them has nn elements. Assume that every two sets from F\mathbb{F} have at most one common element. Prove that i)i) Family F\mathbb{F} has at most n2+nn^2+n elements ii)ii) Upper bound can be reached for n=3n=3
combinatoricsSets
Regional Olympiad - FBH 2010 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
In table of dimensions 2n×2n2n \times 2n there are positive integers not greater than 1010, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least 2n23\frac{2n^2}{3} times
tablecombinatorics
Regional Olympiad - FBH 2010 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
In plane there are nn noncollinear points A1A_1, A2A_2,...,AnA_n. Prove that there exist a line which passes through exactly two of these points
combinatoricsnoncollinear
Regional Olympiad - FBH 2010 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
Let AA1AA_1, BB1BB_1 and CC1CC_1 be altitudes of triangle ABCABC and let A1A2A_1A_2, B1B2B_1B_2 and C1C2C_1C_2 be diameters of Euler circle of triangle ABCABC. Prove that lines AA2AA_2, BB2BB_2 and CC2CC_2 are concurrent
geometryaltitudesEuler Circle