$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left( {\overrightarrow b - \overrightarrow d } \right)\left( {\overrightarrow c - \overrightarrow a } \right) = 0$$$

The point $$D,$$ then, is the ................ of the triangle $$ABC.$$

In a certain city only two newspapers $$A$$ and $$B$$ are published, it is known that $$25$$% of the city population reads $$A$$ and $$20$$% reads $$B$$ while $$8$$% reads both $$A$$ and $$B$$. It is also known that $$30$$% of those who read $$A$$ but not $$B$$ look into advertisements and $$40$$% of those who read $$B$$ but not $$A$$ look into advertisements while $$50$$% of those who read both $$A$$ and $$B$$ look into advertisements. What is the percentage of the population that reads an advertisement?

If 1, $${{a_1}}$$, $${{a_2}}$$......,$${a_{n - 1}}$$ are the n roots of unity, then show that (1- $${{a_1}}$$) (1- $${{a_2}}$$) (1- $${{a_3}}$$) ....$$(1 - \,a{ - _{n - 1}}) = n$$

Evaluate the following $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$