Let $$G\left( x \right) = 1/\left( {1 - x} \right)2 = \sum\limits_{i = 0}^\infty {g\left( i \right)\,{x^1}} \,\,\,,$$
where $$\left| x \right| < 1$$ What is $$g(i)$$?

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $$(a, b)$$ and $$(c, d)$$ in the chosen set such that $$a \equiv c$$ mod $$3$$ and $$b \equiv d$$ mode $$5$$

What is the value of $$\int\limits_0^{2\pi } {{{\left( {x - \pi } \right)}^3}\left( {\sin x} \right)dx} $$

Let $$G$$ be the simple graph with 20 vertices and 100 edges. The size of the minimum vertex cover of $$G$$ is 8. Then, the size of the maximum independent set of $$G$$ is: