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Nužno zlo | Necessary evil #3

Neka su a,b,c > 0 pozitivni realni brojevi koji zadovoljavaju abc = 1. Za svaku od sljedećih tvrdnji odredi je li ona nužno istinita: \begin{enumerate} \item \(\dfrac{ab}{a^5+b^5+ab} + \dfrac{bc}{b^5+c^5+bc} + \dfrac{ac}{a^5+c^5+ac} \geq 1,\) % ne \item \(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \geq a+b+c,\) % da \item \((1+a)(1+b)(1+c) \geq 8,\) % da \item \(8 \geq (a+b)(b+c)(a+c)(a+b-c)(b+c-a)(a+c-b)\), % da \item \(3 \left( \dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a} \right) \geq (a^2+b^2+c^2)^2,\) % ne \item \(a^3+b^3+c^3 + 3 \geq \dfrac{a+b}{c} + \dfrac{b+c}{a} + \dfrac{a+c}{b}\) % 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.

Let a,b,c > 0 be positive real numbers that satisfy abc = 1. For each of the following claims determine whether it necessarily holds true: \begin{enumerate} \item \(\dfrac{ab}{a^5+b^5+ab} + \dfrac{bc}{b^5+c^5+bc} + \dfrac{ac}{a^5+c^5+ac} \geq 1,\) % ne \item \(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \geq a+b+c,\) % da \item \((1+a)(1+b)(1+c) \geq 8,\) % da \item \(8 \geq (a+b)(b+c)(a+c)(a+b-c)(b+c-a)(a+c-b)\), % da \item \(3 \left( \dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a} \right) \geq (a^2+b^2+c^2)^2,\) % ne \item \(a^3+b^3+c^3 + 3 \geq \dfrac{a+b}{c} + \dfrac{b+c}{a} + \dfrac{a+c}{b}\) % 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.