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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!

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  • 1
    $\begingroup$ How did you come up with this inequality? $\endgroup$ – Dimenein Jan 9 '16 at 17:41
  • $\begingroup$ Mathcad graph suggests that the inequality is true. $\endgroup$ – Alex Ravsky Nov 4 '17 at 14:21
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your inequality isn't true, try $a=4,b=3,c=5$

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