Suppose $$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$ If
$$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$$ for all $$x \ge 0$$, prove that
$$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$$.

Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that $$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$$

If $${x^2} + {y^2} = 1$$ then

Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be any point on $${C_2}$$. Let $${P_1}$$ and $${Q_1}$$ be the reflections of $$P$$ and $$Q$$, respectively, with respect to the line $$y=x$$. Prove that $${P_1}$$ lies on $${C_2}$$, $${Q_1}$$ lies on $${C_1}$$ and $$PQ \ge $$ min $$\left\{ {P{P_1},Q{Q_1}} \right\}$$. Hence or otherwise determine points $${P_0}$$ and $${Q_0}$$ on the parabolas $${C_1}$$ and $${C_2}$$ respectively such that $${P_0}{Q_0} \le PQ$$ for all pairs of points $$(P,Q)$$ with $$P$$ on $${C_1}$$ and $$Q$$ on $${C_2}$$.