Let $${z_1}$$ and $${z_2}$$ be roots of the equation $${z^2} + pz + q = 0\,$$ , where the coefficients p and q may be complex numbers. Let A and B represent $${z_1}$$ and $${z_2}$$ in the complex plane. If $$\angle AOB = \alpha \ne 0\,$$ and OA = OB, where O is the origin, prove that $${p^2} = 4q\,{\cos ^2}\left( {{\alpha \over 2}} \right)$$.

Prove that the values of the function $${{\sin x\cos 3x} \over {\sin 3x\cos x}}$$ do not lie between $${1 \over 3}$$ and 3 for any real $$x.$$

Prove that $$\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},} $$ where $$n \ge 3$$ is an integer.

Let $$S$$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $$S$$. If $$a,\,b,\,c$$ and $$d$$ denote the lengths of the sides of the quadrilateral, prove that $$2 \le {a^2} + {b^2} + {c^2} + {d^2} \le 4.$$