MathDB

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Part of 2010 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2010 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
For real numbers aa, bb, cc and dd holds: a+b+c+d=0 a+b+c+d=0 a3+b3+c3+d3=0a^3+b^3+c^3+d^3=0 Prove that sum of some two numbers aa, bb, cc and dd is equal to zero
algebraBosnia
Regional Olympiad - FBH 2010 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
Find all real numbers (x,y)(x,y) satisfying the following: x+3x−yx2+y2=3x+\frac{3x-y}{x^2+y^2}=3 y−x+3yx2+y2=0y-\frac{x+3y}{x^2+y^2}=0
algebrasystem of equations
Regional Olympiad - FBH 2010 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
Prove the inequality y2−x22x2+1+z2−y22y2+1+x2−z22z2+1≥0 \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0 where xx, yy and zz are real numbers
algebrainequalities
Regional Olympiad - FBH 2010 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
It is given positive real number aa such that: {1a}={a2}\left\{\frac{1}{a}\right\}=\{a^2\} 2<a2<3 2<a^2<3 Find the value of a12−144aa^{12}-\frac{144}{a}
algebrafrac