MathDB

4

Part of 2009 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2009 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Let CC be a circle with center OO and radius RR. From point AA of circle CC we construct a tangent tt on circle CC. We construct line dd through point OO whch intersects tangent tt in point MM and circle CC in points BB and DD (BB lies between points OO and MM). If AM=R3AM=R\sqrt{3}, prove: a)a) Triangle AMDAMD is isosceles b)b) Circumcenter of AMDAMD lies on circle CC
geometryisoscelestangentcircumcircle
Regional Olympiad - FBH 2009 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
Let xx and yy be positive integers such that x2−1y+1+y2−1x+1\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1} is integer. Prove that numbers x2−1y+1\frac{x^2-1}{y+1} and y2−1x+1\frac{y^2-1}{x+1} are integers
number theoryInteger
Regional Olympiad - FBH 2009 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2009

9/28/2018
What is the minimal value of 2x+1+3y+1+4z+1\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}, if xx, yy and zz are nonnegative real numbers such that x+y+z=4x+y+z=4
algebrainequalities