MathDB

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Part of 2013 Middle European Mathematical Olympiad

Problems(2)

3 variable

Source: Middle European Mathematical Olympiad 2013 I-1

5/17/2014
Let a,b,c a, b, c be positive real numbers such that a+b+c=1a2+1b2+1c2. a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} . Prove that 2(a+b+c)≥7a2b+13+7b2c+13+7c2a+13. 2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } . Find all triples (a,b,c) (a,b,c) for which equality holds.
inequalities proposedinequalities
Functional equation

Source: Middle European Mathematical Olympiad 2013 T-1

5/17/2014
Find all functions f:R→R f : \mathbb{R} \rightarrow \mathbb{R} such that f(xf(x)+2y)=f(x2)+f(y)+x+y−1 f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 holds for all x,y∈R x, y \in \mathbb{R}.
functionalgebra solvedalgebraFunction equations