Neka su
pozitivni realni brojevi. Za svaku od sljedećih tvrdnji odredi je li ona nužno istinita:
![\begin{enumerate}
\item \(\dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a} \geq a+b+c+ \dfrac{9(a-b)^2}{2(a+b+c)}\), % ne
\item \(a^3+b^3+c^3 \geq 3abc + 3|(a-b)(b-c)(a-c)|\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{c+a} > \dfrac{1}{3} \left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} + \dfrac{(\max\{a,b,c\} - \min \{a,b,c\})^2}{a+b+c} \right)\], % da
\item \(a+b+c \geq \sqrt[3]{abc} + 2\sqrt{\dfrac{a^2+b^2+c^2}{3}}\) % ne
\item \(\left( a+\dfrac{b}{2} + \dfrac{c}{3} \right) \left( a+2b+3c \right) \leq \dfrac{4}{3} (a+b+c)^2\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{a+c} \geq \dfrac{1}{3}\left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} \right) + \dfrac{((a-b)(b-c)(c-a))^{\frac{4}{3}}}{(a+b)(b+c)(a+c)}\] % ne
\end{enumerate}](/media/m/e/b/c/ebc67cb722b15703ea042265f09b19aa.png)
Odgovor zapišite kao niz od
elemenata od kojih je svaki
(nije nužno istinita tvrdnja) ili
(nužno istinita tvrdnja). Primjerice
.
Let
be positive real numbers. For each of the following claims determine whether it necessarily holds true:
Write the answer as a sequence of
elements, each of which is either a
(the claim isn't necessarily true) or a
(the claim is necessarily true). E.g.
.
[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 \(\dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a} \geq a+b+c+ \dfrac{9(a-b)^2}{2(a+b+c)}\), % ne
\item \(a^3+b^3+c^3 \geq 3abc + 3|(a-b)(b-c)(a-c)|\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{c+a} > \dfrac{1}{3} \left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} + \dfrac{(\max\{a,b,c\} - \min \{a,b,c\})^2}{a+b+c} \right)\], % da
\item \(a+b+c \geq \sqrt[3]{abc} + 2\sqrt{\dfrac{a^2+b^2+c^2}{3}}\) % ne
\item \(\left( a+\dfrac{b}{2} + \dfrac{c}{3} \right) \left( a+2b+3c \right) \leq \dfrac{4}{3} (a+b+c)^2\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{a+c} \geq \dfrac{1}{3}\left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} \right) + \dfrac{((a-b)(b-c)(c-a))^{\frac{4}{3}}}{(a+b)(b+c)(a+c)}\] % 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 \(1,0,1,0,1,0\).
[/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 \(\dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a} \geq a+b+c+ \dfrac{9(a-b)^2}{2(a+b+c)}\), % ne
\item \(a^3+b^3+c^3 \geq 3abc + 3|(a-b)(b-c)(a-c)|\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{c+a} > \dfrac{1}{3} \left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} + \dfrac{(\max\{a,b,c\} - \min \{a,b,c\})^2}{a+b+c} \right)\], % da
\item \(a+b+c \geq \sqrt[3]{abc} + 2\sqrt{\dfrac{a^2+b^2+c^2}{3}}\) % ne
\item \(\left( a+\dfrac{b}{2} + \dfrac{c}{3} \right) \left( a+2b+3c \right) \leq \dfrac{4}{3} (a+b+c)^2\), % da
\item Ako je \(a \neq b \neq c \neq a\), tada
\[\dfrac{a^2}{a+b} + \dfrac{b^2}{b+c} + \dfrac{c^2}{a+c} \geq \dfrac{1}{3}\left( \dfrac{a^3-b^3}{a^2-b^2} + \dfrac{b^3-c^3}{b^2-c^2} + \dfrac{c^3-a^3}{c^2-a^2} \right) + \dfrac{((a-b)(b-c)(c-a))^{\frac{4}{3}}}{(a+b)(b+c)(a+c)}\] % 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. \(1,0,1,0,1,0\).
[/lang]
