MathDB

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Part of 2012 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2012 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
Find all possible values of 1a(1b+1c+1b+c)+1b(1c+1a+1c+a)+1c(1a+1b+1a+b)−1a+b+c(1a+1b+1c+1a+b+1b+c+1c+a)+1a2+1b2+1c2\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} where aa, bb and cc are positive real numbers such that ab+bc+ca=abcab+bc+ca=abc
algebraidentityvalue
Regional Olympiad - FBH 2012 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
Solve equation x2−a−x=ax^2-\sqrt{a-x}=a where xx is real number and aa is real parameter
equationparameterizationalgebra
Regional Olympiad - FBH 2012 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
For which real numbers xx and α\alpha inequality holds: log⁡2x+log⁡x2+2cos⁡α≤0\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0
inequalitiesalgebra