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 $$2 \le {a^2} + {b^2} + {c^2} + {d^2} \le 4.$$

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 $$x.$$

The sum of all the real roots of the equation $${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$$ is ............................

The sum of the rational terms in the expansion of $${\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}$$ is ...............