Marinada '22

[lang=hr] Neka su \(a,b,c > 0\) pozitivni realni brojevi. Za svaku od sljedećih tvrdnji odredi je li ona nužno istinita: \begin{enumerate} \item \(a^3+b^3+c^3 \geq 3abc,\) % da \item \(a^2+b^2+c^2 \geq a+b+c,\) % ne \item \(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \geq a+b+c\), % ne \item \(\dfrac{a^2}{b^2} + \dfrac{b^2}{c^2} + \dfrac{c^2}{a^2} \geq \dfrac{a}{c} + \dfrac{b}{a} + \dfrac{c}{b},\) % da \item \(\dfrac{a}{2b^3+1} + \dfrac{b}{2c^3+1} + \dfrac{c}{2a^3+1} \geq 1\), % ne \item \((ab+bc+ac)^2 \geq 3abc(a+b+c)\) % da \end{enumerate} Odgovor zapišite kao niz od \(6\) elemenata od kojih je svaki \(0\) (nije nužno istinita tvrdnja) ili \(1\) (nužno istinita tvrdnja). Primjerice \(1,0,1,0,1,0\). [/lang] [lang=en] Let \(a,b,c > 0\) be positive real numbers. For each of the following claims determine whether it necessarily holds true: \begin{enumerate} \item \(a^3+b^3+c^3 \geq 3abc,\) % da \item \(a^2+b^2+c^2 \geq a+b+c,\) % ne \item \(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \geq a+b+c\), % ne \item \(\dfrac{a^2}{b^2} + \dfrac{b^2}{c^2} + \dfrac{c^2}{a^2} \geq \dfrac{a}{c} + \dfrac{b}{a} + \dfrac{c}{b},\) % da \item \(\dfrac{a}{2b^3+1} + \dfrac{b}{2c^3+1} + \dfrac{c}{2a^3+1} \geq 1\), % ne \item \((ab+bc+ac)^2 \geq 3abc(a+b+c)\) % da \end{enumerate} Write the answer as a sequence of \(6\) elements, each of which is either a \(0\) (the claim isn't necessarily true) or a \(1\) (the claim is necessarily true). E.g. \(1,0,1,0,1,0\). [/lang]