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Inequality in a triangle

General Mathematics
Riemann
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Inequality in a triangle

Let $ABC$ be a triangle and denote $a, b, c$ the lengths of the sides $BC , CA$ and $AB$ respectively. If $abc \geq 1$ then prove that

$$\sqrt{\frac{\sin A}{a^3+b^6+c^6}} + \sqrt{\frac{\sin B}{b^3+c^6+a^6}} + \sqrt{\frac{\sin C}{c^3 + a^6+b^6}} \leq \sqrt[4]{\frac{27}{4}}$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$