<s>[INST] Howto prove that $\sum\limits_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$ let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that
$$\cos\frac{A}{2}\cos\frac{B}{2}+\cos\frac{C}{2}\cos\frac{B}{2}+\cos\frac{A}{2}\cos\frac{C}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$$
This problem is created by me. I believe that this is true, but I can't prove it.
This problem is motivated by a 1988 IMO Longlists problem
$$ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. $$
I post the solution:
\begin{align*} & 4\sum \sin \frac A2 \sin \frac B2 \leq \frac 32 + 1 + \frac {r}{R} \\
\iff & 4\sum \sin \frac A2 \sin \frac B2 \leq \frac 32+ \sum \cos A \\
\iff & 4\sum \sin \frac A2 \sin \frac B2 \leq \frac 32 + \sum (1 - 2\sin^2 \frac A2) \\
\iff & 4\sum \sin \frac A2 \sin \frac B2 + 2\sum \sin^2 \frac A2 \leq \frac 92 \\
\iff & 2\left(\sum\sin\frac {A}{2}\right)^{2} \leq \frac 92 \\
\iff & \left(\sum\sin\frac {A}{2}\right)^{2}\leq \frac 94 \end{align*}
The last inequality is a well know inequality. We are done!
But My inequality is stronger this IMOLonglists problem, so I can't prove it. Thank you for you help!
 [/INST] your inequality isn't true, try $a=4,b=3,c=5$
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