# Inequality

Algebra Level 5

Given

$\begin{cases}a,b,c,d>0\\ab+bc+cd+da=1,\end{cases}$

$k$ is the largest number such that

$\displaystyle \frac{a^3+c^3}{2\sqrt{bd}}+\frac{b^3+d^3}{2\sqrt{ac}}\ge k.$

Equality is reached when $a=a_{\text{eq}}$, $b=b_{\text{eq}}$, $c=c_{\text{eq}}$, $d=d_{\text{eq}}$.

Find $a_{\text{eq}}+b_{\text{eq}}+c_{\text{eq}}+d_{\text{eq}}+k$ and round it to $3$ decimal places.

Original.

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