Vrijeme: 08:15

Nužno zlo | Necessary evil #2

Neka su a,b,c > 0 pozitivni realni brojevi koji zadovoljavaju a+b+c = 3. Za svaku od sljedećih tvrdnji odredi je li ona nužno istinita: \begin{enumerate} \item \(\sqrt{1+a} + \sqrt{1+b} + \sqrt{1+c} \geq 3\sqrt{2},\) % ne \item \(\dfrac{a}{3-a} + \dfrac{b}{3-b} + \dfrac{c}{3-c} \geq \dfrac{3}{2},\) % da \item \((a+b)(b+c)(a+c) \geq \dfrac{8}{3}(ab+bc+ac),\) % da \item \(a^2+b^2+c^2 \geq a+b+c\), % da \item \(a^3b + b^3c + c^3a \geq a^2b^2 + b^2c^2 + a^2c^2\), % ne \item \(a^2b^2 + b^2c^2 + a^2c^2 \geq \sqrt{a} + \sqrt{b} + \sqrt{c}\) % ne \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 a+b+c = 3. For each of the following claims determine whether it necessarily holds true: \begin{enumerate} \item \(\sqrt{1+a} + \sqrt{1+b} + \sqrt{1+c} \geq 3\sqrt{2},\) % ne \item \(\dfrac{a}{3-a} + \dfrac{b}{3-b} + \dfrac{c}{3-c} \geq \dfrac{3}{2},\) % da \item \((a+b)(b+c)(a+c) \geq \dfrac{8}{3}(ab+bc+ac),\) % da \item \(a^2+b^2+c^2 \geq a+b+c\), % da \item \(a^3b + b^3c + c^3a \geq a^2b^2 + b^2c^2 + a^2c^2\), % ne \item \(a^2b^2 + b^2c^2 + a^2c^2 \geq \sqrt{a} + \sqrt{b} + \sqrt{c}\) % ne \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.