MathDB

4

Part of 2017 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2017 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
It is given isosceles triangle ABCABC (AB=ACAB=AC) such that ∠BAC=108∘\angle BAC=108^{\circ}. Angle bisector of angle ∠ABC\angle ABC intersects side ACAC in point DD, and point EE is on side BCBC such that BE=AEBE=AE. If AE=mAE=m, find EDED
geometryangle bisectorisosceles
Regional Olympiad - FBH 2017 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
Let SS be a set of nn distinct real numbers, and ASA_S set of arithemtic means of two distinct numbers from SS. For given n≥2n \geq 2 find minimal number of elements in ASA_S
arithmetic meanSetsnumber theorycombinatorics
Regional Olympiad - FBH 2017 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
It is given positive integer NN. Let d1d_1, d2d_2,...,dnd_n be its divisors and let aia_i be number of divisors of did_i, i=1,2,...ni=1,2,...n. Prove that (a1+a2+...+an)2=a13+a23+...+an3(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3
Divisorsnumber theorynumber of divisors
Regional Olympiad - FBH 2017 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
How many knights you can put on chess table 5×55 \times 5 such that every one of them attacks exactly two other knights ?
chessknightsarrangingcombinatorics