MathDB

1

Part of 2009 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2009 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Find all triplets of integers (x,y,z)(x,y,z) such that xy(x2−y2)+yz(y2−z2)+zx(z2−x2)=1xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1
number theoryIntegers
Regional Olympiad - FBH 2009 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
In triangle ABCABC such that ∠ACB=90∘\angle ACB=90^{\circ}, let point HH be foot of perpendicular from point CC to side ABAB. Show that sum of radiuses of incircles of ABCABC, BCHBCH and ACHACH is CHCH
geometryincircleright angle
Regional Olympiad - FBH 2009 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
In triangle ABCABC holds ∠ACB=90∘\angle ACB = 90^{\circ}, ∠BAC=30∘\angle BAC = 30^{\circ} and BC=1BC=1. In triangle ABCABC is inscribed equilateral triangle (every side of a triangle ABCABC contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle
geometryinscribed triangle
Regional Olympiad - FBH 2009 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Prove that for every positive integer mm there exists positive integer nn such that m+n+1m+n+1 is perfect square and mn+1mn+1 is perfect cube of some positive integers
number theoryperfect cubePerfect Square