The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues.
Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm?
A) 32
B) 36
C) 38
D) 40
E) 48

An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles.
What is the sum of the areas of all nested triangular regions drawn in this manner, in square units?
A) $\frac { \sqrt { 3 } } { 3 }$
B) $\frac { 3 \sqrt { 3 } } { 4 }$
C) $\frac { 8 \sqrt { 3 } } { 9 }$
D) $\frac { 5 \sqrt { 3 } } { 16 }$
E) $\frac { 9 \sqrt { 3 } } { 32 }$

Below, a sequence of circles drawn side by side is given. In this sequence; the radius of the first circle is 4 units and the radius of each subsequent circle is half the radius of the previous circle.
What is the sum of the circumferences of all circles in this sequence in units?
A) $15 \pi$
B) $16 \pi$
C) $18 \pi$
D) $\frac { 31 \pi } { 2 }$
E) $\frac { 33 \pi } { 2 }$

The first three terms of a geometric sequence are $\mathbf { a } + \mathbf { 3 }$, a, and $\mathbf { a } - \mathbf { 2 }$ respectively. Accordingly, what is the fourth term?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 6 }$

An equilateral triangle is inscribed in circle $\mathrm { C } _ { 1 }$ with radius 1 unit as shown in the figure. Let $\mathrm { C } _ { 2 }$ be the circle passing through the midpoints of the sides of this triangle. The same operation is performed for circle $\mathrm { C } _ { 2 }$ and circle $\mathrm { C } _ { 3 }$ is obtained. By repeating this process infinitely many times, the sequence of circles $C _ { 1 } , C _ { 2 } , C _ { 3 } , \cdots$ is obtained.
For each positive integer $\mathbf { k }$, let $\mathbf { A } _ { \mathbf { k } }$ be the area bounded by circle $\mathbf { C } _ { \mathbf { k } }$ in square units. Accordingly, what is the result of the sum
$$\sum _ { k = 1 } ^ { \infty } A _ { k }$$
?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\frac { 6 \pi } { 5 }$
E) $\frac { 9 \pi } { 8 }$

In the rectangular coordinate plane, isosceles triangles are drawn with base vertices at consecutive even natural numbers on the x-axis and apex on the curve $y = 2 ^ { - x }$.
Accordingly, what is the sum of the areas of all the triangles drawn in square units?
A) $\frac { 3 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 1
E) 2

Let $(a_n)$ be a geometric sequence. The equality
$$\frac { a _ { 5 } - a _ { 1 } } { \left( a _ { 3 } \right) ^ { 2 } - \left( a _ { 1 } \right) ^ { 2 } } = \frac { 4 } { 9 }$$
is given. Given that $a _ { 2 } = \frac { 3 } { 2 }$, what is $a _ { 4 }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 27 } { 8 }$
E) $\frac { 27 } { 4 }$

In the first quadrant of the rectangular coordinate plane; a square $A _ { 1 }$ is drawn with two sides on the coordinate axes and one vertex on the line $\mathrm { d } : \mathrm { y } = 4 - \mathrm { x }$. Then, a square $A _ { 2 }$ adjacent to the square $A _ { 1 }$ with one side on the x-axis and one vertex on line d is drawn. Continuing in a similar manner, a sequence of squares $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 } , \mathrm {~A} _ { 3 } , \ldots$ is obtained as shown in the figure. Accordingly, what is the sum of the areas of all the squares $\mathbf { A } _ { \mathbf { n } }$ obtained in square units?
A) $\frac { 9 } { 2 }$
B) $\frac { 11 } { 2 }$
C) $\frac { 14 } { 3 }$
D) $\frac { 16 } { 3 }$
E) $\frac { 20 } { 3 }$

For a geometric sequence $(a_n)$ with all positive terms and common ratio $r$
$$\begin{aligned} & a_1 + \frac{1}{2} + r \\ & a_7^2 = a_5 + 12 \cdot a_3 \end{aligned}$$
the equalities are given.