Consider an equilateral triangle $ABC$ with altitude 3 centimeters. A circle is inscribed in this triangle, then another circle is drawn such that it is tangent to the inscribed circle and the sides $AB$, $AC$. Infinitely many such circles are drawn; each tangent to the previous circle and the sides $AB$, $AC$. Find the sum of the areas of all these circles.

As shown in the figure below, from a square with side length 1, a square with side length $\frac { 1 } { 2 }$ is cut out, and the remaining shape is called $A _ { 1 }$. From a square with side length $\frac { 1 } { 4 }$, a square with side length $\frac { 1 } { 8 }$ is cut out, and two resulting shapes are attached to the upper two sides of $A _ { 1 }$ to form a figure called $A _ { 2 }$. From a square with side length $\frac { 1 } { 16 }$, a square with side length $\frac { 1 } { 32 }$ is cut out, and four resulting shapes are attached to the upper four sides of $A _ { 2 }$ to form a figure called $A _ { 3 }$.
Continuing this process, let $A _ { n }$ be the $n$-th figure obtained and $S _ { n }$ be its area. If $\lim _ { n \rightarrow \infty } S _ { n } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

There is a right isosceles triangle with the two legs forming the right angle each having length 1. A square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring this square is called $R _ { 1 }$. In figure $R _ { 1 }$, 2 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 2 squares is called $R _ { 2 }$. In figure $R _ { 2 }$, 4 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 4 squares is called $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 3 \sqrt { 2 } } { 20 }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { \sqrt { 3 } } { 5 }$
(5) $\frac { 2 } { 5 }$

Inside a rectangle with width 6 and height 8, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle is drawn to obtain figure $R _ { 1 }$. From figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 2 }$.
In figure $R _ { 2 }$, for each of the four congruent rectangles, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the widths and heights of all rectangles are parallel to each other, respectively.) [4 points]
(1) $\frac { 37 } { 9 } \pi$
(2) $\frac { 34 } { 9 } \pi$
(3) $\frac { 31 } { 9 } \pi$
(4) $\frac { 28 } { 9 } \pi$
(5) $\frac { 25 } { 9 } \pi$

On the coordinate plane, there is a circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + y ^ { 2 } = 1$. As shown in the figure, a tangent line $l$ with positive slope is drawn from the origin to the circle $C _ { 1 }$, and the point of tangency is called $\mathrm { P } _ { 1 }$.
A circle $C _ { 2 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 1 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 2 }$ be the point of tangency between this circle and the $x$-axis.
A circle $C _ { 3 }$ has its center on the $x$-axis, passes through the point $\mathrm { P } _ { 2 }$, and is tangent to the line $l$. Let $\mathrm { P } _ { 3 }$ be the point of tangency between this circle and the line $l$.
A circle $C _ { 4 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 3 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 4 }$ be the point of tangency between this circle and the $x$-axis. Continuing this process, let $S _ { n }$ be the area of circle $C _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? (Note: The radius of circle $C _ { n + 1 }$ is smaller than the radius of circle $C _ { n }$.) [4 points]
(1) $\frac { 3 } { 2 } \pi$
(2) $2 \pi$
(3) $\frac { 5 } { 2 } \pi$
(4) $3 \pi$
(5) $\frac { 7 } { 2 } \pi$

There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, determine two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$, and determine two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, determine two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process to obtain $S _ { n }$, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$

There is a circle with radius 1. As shown in the figure, a rectangle with the ratio of width to height of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$

There is a circle with radius 1. As shown in the figure, a rectangle with a ratio of horizontal length to vertical length of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain a figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangles. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$

As shown in the figure, there is a circle O with diameter AB of length 2. Let C be one of the two points where the line passing through the center of circle O and perpendicular to segment AB meets the circle. The figure obtained by shading the region that is outside the circle centered at C passing through points A and B and inside circle O is called $R_1$. In figure $R_1$, circles are inscribed in each of the 2 quarter-circles obtained by bisecting the semicircle of circle O that does not include the shaded part. In these 2 circles, 2 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_2$. In figure $R_2$, circles are inscribed in each of the 4 quarter-circles obtained by bisecting the semicircles of the 2 newly created circles that do not include the shaded parts. In these 4 circles, 4 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_3$. Continuing this process, let $S_n$ be the area of the shaded part in the $n$-th figure $R_n$. What is the value of $\lim_{n \rightarrow \infty} S_n$? [4 points]
(1) $\frac{5 + 2\sqrt{2}}{7}$
(2) $\frac{5 + 3\sqrt{2}}{7}$
(3) $\frac{5 + 4\sqrt{2}}{7}$
(4) $\frac{5 + 5\sqrt{2}}{7}$
(5) $\frac{5 + 6\sqrt{2}}{7}$

As shown in the figure, there is a circle O with diameter AB of length 2. A line passing through the center of circle O and perpendicular to line segment AB intersects the circle at two points, one of which is C.
A circle centered at C passing through points A and B is drawn. The region that is outside this circle and inside circle O is colored to form a triangular shape, creating figure $R _ { 1 }$. The semicircle of circle O that does not include the colored part is divided into 2 quarter circles, and circles inscribed in each quarter circle are drawn. Inside these 2 circles, two triangular shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 2 }$. The semicircles of the 2 newly created circles in figure $R _ { 2 }$ that do not include the colored parts are each divided into 2 quarter circles, and circles inscribed in each of the 4 quarter circles are drawn. Inside these 4 circles, 4 shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 + 2 \sqrt { 2 } } { 7 }$
(2) $\frac { 5 + 3 \sqrt { 2 } } { 7 }$
(3) $\frac { 5 + 4 \sqrt { 2 } } { 7 }$
(4) $\frac { 5 + 5 \sqrt { 2 } } { 7 }$
(5) $\frac { 5 + 6 \sqrt { 2 } } { 7 }$

As shown in the figure, let $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ be the five equal division points of the diagonal BD of a square ABCD with side length 5, in order from point B. Draw squares with line segments $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ as diagonals respectively, and circles with line segments $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$ as diameters respectively. Color the figure-eight-shaped regions to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ that is closest to point A, and let $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw a square with diagonal $\mathrm { AQ } _ { 1 }$ and a square with diagonal $\mathrm { CQ } _ { 2 }$, and in these 2 newly drawn squares, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 1 }$, and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the square with diagonal $\mathrm { AQ } _ { 1 }$ and the square with diagonal $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 2 }$ from figure $R _ { 1 }$, and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$

As shown in the figure, there is a right triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ with $\overline { \mathrm { OA } _ { 1 } } = 4$ and $\overline { \mathrm { OB } _ { 1 } } = 4 \sqrt { 3 }$. Let $\mathrm { B } _ { 2 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 1 } }$ meets the line segment $\mathrm { OB } _ { 1 }$. The figure $R _ { 1 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ but outside the sector $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 2 }$. In figure $R _ { 1 }$, let $\mathrm { A } _ { 2 }$ be the point where the line passing through $\mathrm { B } _ { 2 }$ and parallel to segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$ meets segment $\mathrm { OA } _ { 1 }$, and let $\mathrm { B } _ { 3 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 2 } }$ meets segment $\mathrm { OB } _ { 2 }$. The figure $R _ { 2 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 2 }$ but outside the sector $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points] [Figure] [Figure]
(1) $\frac { 3 } { 2 } \pi$
(2) $\frac { 5 } { 3 } \pi$
(3) $\frac { 11 } { 6 } \pi$
(4) $2 \pi$
(5) $\frac { 13 } { 6 } \pi$

As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$. In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 441 } { 103 }$
(2) $\frac { 441 } { 109 }$
(3) $\frac { 441 } { 115 }$
(4) $\frac { 441 } { 121 }$
(5) $\frac { 441 } { 127 }$

Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then:
(1) $\mathrm { P } ^ { 2 } = 6 \sqrt { 3 } \mathrm { Q }$
(2) $\mathrm { P } ^ { 2 } = 36 \sqrt { 3 } \mathrm { Q }$
(3) $P = 36 \sqrt { 3 } Q ^ { 2 }$
(4) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$

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) $\frac { 33 \pi } { 2 }$

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

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) $\frac { 20 } { 3 }$