MathDB

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Part of 2013 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2013 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
If xx and yy are real numbers such that x2013+y2013>x2012+y2012x^{2013}+y^{2013}>x^{2012}+y^{2012}, prove that x2014+y2014>x2013+y2013x^{2014}+y^{2014}>x^{2013}+y^{2013}
algebrainequalities
Regional Olympiad - FBH 2013 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
If xx and yy are nonnegative real numbers such that x+y=1x+y=1, determine minimal and maximal value of A=x1+y+y1+xA=x\sqrt{1+y}+y\sqrt{1+x}
inequalitiesminimummaximum
Regional Olympiad - FBH 2013 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
Let aa and bb be real numbers from interval [0,π2]\left[0,\frac{\pi}{2}\right]. Prove that sin⁡6a+3sin⁡2acos⁡2b+cos⁡6b=1\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1 if and only if a=ba=b
algebraequationtrigonometry
Regional Olympiad - FBH 2013 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
If aa, bb and cc are nonnegative real numbers such that a2+b2+c2=1a^2+b^2+c^2=1, prove that 12≤a1+a4+b1+b4+c1+c4≤9310\frac{1}{2} \leq \frac{a}{1+a^4}+\frac{b}{1+b^4}+\frac{c}{1+c^4} \leq \frac{9\sqrt{3}}{10}
algebraInequality