Four numbers $1 , a , b , c$ form a geometric sequence with common ratio $r$ in this order, and satisfy $\log _ { 8 } c = \log _ { a } b$. What is the value of the common ratio $r$? (where $r > 1$) [3 points]
(1) 2
(2) $\frac { 5 } { 2 }$
(3) 3
(4) $\frac { 7 } { 2 }$
(5) 4

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) $\frac { 7 } { 2 } \pi$

For two infinite geometric sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ with the same common ratio, $a _ { 1 } - b _ { 1 } = 1$, $\sum _ { n = 1 } ^ { \infty } a _ { n } = 8$, and $\sum _ { n = 1 } ^ { \infty } b _ { n } = 6$. Find the value of $\sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }$. [3 points]

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } = \frac { 1 } { 2 }$ and $a _ { 5 } = \frac { 1 } { 6 }$. When $\sum _ { n = 1 } ^ { \infty } a _ { n } a _ { n + 1 } a _ { n + 2 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For two natural numbers $a$ and $b$, the three numbers $a ^ { n } , 2 ^ { 4 } \times 3 ^ { 6 } , b ^ { n }$ form a geometric sequence in this order. Find the minimum value of $a b$. (Here, $n$ is a natural number.) [4 points]

When $\lim _ { n \rightarrow \infty } \frac { a \times 6 ^ { n + 1 } - 5 ^ { n } } { 6 ^ { n } + 5 ^ { n } } = 4$, what is the value of the constant $a$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 3 } { 2 }$

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) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$

On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$. Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 1 } a _ { 2 } } { a _ { 3 } } = 2 , \quad \frac { 2a _ { 2 } } { a _ { 1 } } + \frac { a _ { 4 } } { a _ { 2 } } = 8$$ what is the value of $a _ { 3 }$? [3 points]
(1) 16
(2) 18
(3) 20
(4) 22
(5) 24

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) $\frac { 5 + 6 \sqrt { 2 } } { 7 }$

For a geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio, if $a _ { 1 } = 3 , a _ { 5 } = 48$, what is the value of $a _ { 3 }$? [3 points]
(1) 18
(2) 16
(3) 14
(4) 12
(5) 10

For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 89 } { 8 }$
(2) $\frac { 87 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 83 } { 8 }$
(5) $\frac { 81 } { 8 }$

For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 81 } { 8 }$
(2) $\frac { 83 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 87 } { 8 }$
(5) $\frac { 89 } { 8 }$

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { n = 1 } ^ { \infty } a _ { n } = 4 , \quad \sum _ { n = 1 } ^ { \infty } b _ { n } = 10$$ find the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + 5 b _ { n } \right)$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with a non-zero first term, $$a _ { 3 } = 4 a _ { 1 } , \quad a _ { 7 } = \left( a _ { 6 } \right) ^ { 2 }$$ what is the value of the first term $a _ { 1 }$? [3 points]
(1) $\frac { 1 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 5 } { 16 }$

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) $\frac { 25 } { 17 } ( 2 \pi + 1 )$

Find the value of $\lim _ { n \rightarrow \infty } \frac { 3 \times 9 ^ { n } - 13 } { 9 ^ { n } }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 1 and common ratio $r$ ($r > 1$), let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. When $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { S _ { n } } = \frac { 3 } { 4 }$, find the value of $r$. [3 points]

When three numbers $\frac { 9 } { 4 } , a , 4$ form a geometric sequence in this order, what is the value of the positive number $a$? [3 points]
(1) $\frac { 8 } { 3 }$
(2) 3
(3) $\frac { 10 } { 3 }$
(4) $\frac { 11 } { 3 }$
(5) 4

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) $\frac { 13 } { 6 } \pi$

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 7, let $S _ { n }$ denote the sum of the first $n$ terms. $$\frac { S _ { 9 } - S _ { 5 } } { S _ { 6 } - S _ { 2 } } = 3$$ Find the value of $a _ { 7 }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 16 } } { a _ { 14 } } + \frac { a _ { 8 } } { a _ { 7 } } = 12$$ Find the value of $\frac { a _ { 3 } } { a _ { 1 } } + \frac { a _ { 6 } } { a _ { 3 } }$. [3 points]