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

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 ...............

Let $$0 < {A_i} < n$$ for $$i = 1,\,2....,\,n.$$ Use mathematical induction to prove that $$$\sin {A_1} + \sin {A_2}....... + \sin {A_n} \le n\,\sin \,\,\left( {{{{A_1} + {A_2} + ...... + {A_n}} \over n}} \right)$$$

where $$ \ge 1$$ is a natural number. {You may use the fact that $$p\sin x + \left( {1 - p} \right)\sin y \le \sin \left[ {px + \left( {1 - p} \right)y} \right],$$ where $$0 \le p \le 1$$ and $$0 \le x,y \le \pi .$$}