Old Tor Vergata
Inviato: 12 nov 2020, 09:14
Trovare $n$ intero positivo in modo che per ogni quaterna di numeri reali $(x,y,w,z)$ non tutti nulli, si abbia: $$\frac {x^{35}y^{15}z^{24}w^n}{x^{84}+y^{180}+z^{120}+w^{440}} \leq 2012$$
Testo nascosto:
Applichiamo AM-GM
\begin{equation}
25 \cdot \frac{x^{84}}{25} + 5 \cdot \frac{y^{180}}{5} + 12 \cdot \frac{z^{120}}{12} + 18 \cdot \frac{w^{440}}{18} \geq 60 \cdot \sqrt[60]{\left(\frac{x^{84}}{25}\right)^{25}\left(\frac{y^{180}}{5}\right)^{5}\left(\frac{z^{120}}{12}\right)^{12}\left(\frac{w^{440}}{18}\right)^{18}} = 60 \cdot \frac{x^{35}y^{15}z^{24}w^{132}}{\sqrt[60]{5^{55} \cdot 3^{48} \cdot 2^{42}}}
\end{equation}
Quindi abbiamo
\begin{equation}
\frac{x^{35}y^{15}z^{24}w^{132}}{x^{84}+y^{180}+z^{120}+w^{440}} \leq \frac{5 \cdot 3 \cdot 2}{60 \cdot \sqrt[60]{5^{5} \cdot 3^{12} \cdot 2^{18}}} = \frac{1}{2 \cdot \sqrt[12]{5} \cdot \sqrt[5]{3} \cdot \sqrt[10]{8}} \leq 2012
\end{equation}
\begin{equation}
\Rightarrow n=132
\end{equation}
\begin{equation}
25 \cdot \frac{x^{84}}{25} + 5 \cdot \frac{y^{180}}{5} + 12 \cdot \frac{z^{120}}{12} + 18 \cdot \frac{w^{440}}{18} \geq 60 \cdot \sqrt[60]{\left(\frac{x^{84}}{25}\right)^{25}\left(\frac{y^{180}}{5}\right)^{5}\left(\frac{z^{120}}{12}\right)^{12}\left(\frac{w^{440}}{18}\right)^{18}} = 60 \cdot \frac{x^{35}y^{15}z^{24}w^{132}}{\sqrt[60]{5^{55} \cdot 3^{48} \cdot 2^{42}}}
\end{equation}
Quindi abbiamo
\begin{equation}
\frac{x^{35}y^{15}z^{24}w^{132}}{x^{84}+y^{180}+z^{120}+w^{440}} \leq \frac{5 \cdot 3 \cdot 2}{60 \cdot \sqrt[60]{5^{5} \cdot 3^{12} \cdot 2^{18}}} = \frac{1}{2 \cdot \sqrt[12]{5} \cdot \sqrt[5]{3} \cdot \sqrt[10]{8}} \leq 2012
\end{equation}
\begin{equation}
\Rightarrow n=132
\end{equation}