As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]

For a triangle $ABC$ with side lengths $|BC| = a$ units, $|AC| = b$ units and $|AB| = c$ units,
$$2a^{2} = 2b^{2} + 2c^{2} + 3bc$$
is satisfied. Let $m(\widehat{BAC}) = x$. What is the value of $\tan x$?
A) $-\frac{\sqrt{2}}{3}$ B) $-\frac{\sqrt{3}}{3}$ C) $-\frac{\sqrt{5}}{3}$ D) $-\frac{\sqrt{6}}{3}$ E) $-\frac{\sqrt{7}}{3}$

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$