CMO 1969

If $a_1/b_1=a_2/b_2=a_3/b_3$ and $p_1,p_2,p_3$ are not all zero, show that for all $n\in\mathbb{N}$ , $$\left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}.$$

Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, $$\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}.$$

Let $ABC$ be a triangle with sides of length $a$ , $b$ and $c$ . Let the bisector of the angle $C$ cut $AB$ in $D$ . Prove that the length of $CD$ is $$\frac{2ab\cos \frac{C}{2}}{a+b}.$$