MathDB

2

Part of 2009 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2009 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Find minimum of x+y+zx+y+z where xx, yy and zz are real numbers such that x≥4x \geq 4, y≥5y \geq 5, z≥6z \geq 6 and x2+y2+z2≥90x^2+y^2+z^2 \geq 90
minimumalgebrainequalities
Regional Olympiad - FBH 2009 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Find minimal value of a∈Ra \in \mathbb{R} such that system x−1+y−1+z−1=a−1\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1 x+1+y+1+z+1=a+1\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1 has solution in set of real numbers
Systemalgebraminimuminequalities
Regional Olympiad - FBH 2009 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
For given positive integer nn find all quartets (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) such that x12+x22+x32+x42=4nx_1^2+x_2^2+x_3^2+x_4^2=4^n
number theoryquartet
Regional Olympiad - FBH 2009 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Let ABCABC be an equilateral triangle such that length of its altitude is 11. Circle with center on the same side of line ABAB as point CC and radius 11 touches side ABAB. Circle rolls on the side ABAB. While the circle is rolling, it constantly intersects sides ACAC and BCBC. Prove that length of an arc of the circle, which lies inside the triangle, is constant
geometryEquilateral Trianglefixed