
Inequality very hard
Let $\displaystyle a,b,c,d\in \mathbb{R}$ and $\displaystyle \left\{\begin{matrix}
0<a\leq b\leq c\leq d\\
\frac{1}{a}+\frac{2}{b}+\frac{d}{c}\geq 3\\
\frac{2}{b}+\frac{d}{c}\geq 2
\end{matrix}\right.$. Prove that:
$\displaystyle a^4+b^4+c^4d^4\leq 17$

Re: Inequality very hard
Unfortunately I can't help you with your problem, but I did fix your LaTex codes so that someone might have an easier time helping you. In the future, use '\\' (without the apostrophes) to create a line space. Pressing enter will produce these annoying BR tags :)
Let $\displaystyle a,b,c,d\in \mathbb{R}$ and $\displaystyle \left\{\begin{matrix}0<a\leq b\leq c\leq d\\ \frac{1}{a}+\frac{2}{b}+\frac{d}{c}\geq 3\\ \frac{2}{b}+\frac{d}{c}\geq 2 \end{matrix}\right.$. Prove that:
$\displaystyle a^4+b^4+c^4d^4\leq 17$