Inequality

Algebra Level 5

Given

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

kk is the largest number such that

a3+c32bd+b3+d32ack.\displaystyle \frac{a^3+c^3}{2\sqrt{bd}}+\frac{b^3+d^3}{2\sqrt{ac}}\ge k.

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

Find aeq+beq+ceq+deq+ka_{\text{eq}}+b_{\text{eq}}+c_{\text{eq}}+d_{\text{eq}}+k and round it to 33 decimal places.


Original.

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