For a triangle $$ABC$$ it is given that $$\cos A + \cos B + \cos C = {3 \over 2}$$. Prove that the triangle is equilateral.

With usual notation, if in a triangle $$ABC$$;
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.

The numerical value of $$\tan \left\{ {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) - {\pi \over 4}} \right\}$$ is equal to __________

For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$.