Školjka

%V0 $\dfrac{\cos\alpha}{a^3} + \dfrac{\cos\beta}{b^3} + \dfrac{\cos\gamma}{c^3} \geq \dfrac{3}{2abc} \Leftrightarrow$ $\dfrac{2bc\cos\alpha}{a^2} + \dfrac{2ac\cos\beta}{b^2} + \dfrac{2ab\cos\gamma}{c^2} \geq 3 \Leftrightarrow$ $\dfrac{b^2+c^2-a^2}{a^2} + \dfrac{a^2+c^2-b^2}{b^2} + \dfrac{a^2+b^2-c^2}{c^2} \geq 3 \Leftrightarrow$ ( iz kosinusovog poučka ) $ \left(\dfrac{a^2}{b^2} + \dfrac{b^2}{a^2}\right) + \left(\dfrac{c^2}{a^2} + \dfrac{a^2}{c^2}\right) + \left(\dfrac{b^2}{c^2} + \dfrac{c^2}{b^2}\right) - 3 \geq 3$ No vrijedi i $\dfrac{x}{y} + \dfrac{y}{x} \geq 2$, te stoga imamo: $ LHS \geq 6 -3 = 3 $, čime je početna nejednakost dokazana. $\blacksquare$