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

A graph and the matrix representing the connection relationships between each vertex of the graph are as follows. What is the value of $a + b + c + d + e$?
A B C D E
$$\left( \begin{array} { l l l l l } 0 & 1 & 1 & 0 & a \\ 1 & 0 & 1 & b & 1 \\ 1 & 1 & c & 1 & 0 \\ 0 & d & 1 & 0 & 1 \\ e & 1 & 0 & 1 & 0 \end{array} \right)$$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds:
$$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$
(Here, $k$ is a positive constant, the unit of length is m, and the unit of speed is m/s.)
When the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center. What is the value of $a$? [3 points]
(1) $\frac { 39 } { 23 }$
(2) $\frac { 37 } { 23 }$
(3) $\frac { 35 } { 23 }$
(4) $\frac { 33 } { 23 }$
(5) $\frac { 31 } { 23 }$

The graph of the function $y = f ( x )$ is shown in the figure.
What is the value of $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1$ and $\overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ and $\mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$. The region bounded by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region bounded by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ form a checkmark shape. Color this shape to obtain figure $R _ { 1 }$.
In figure $R _ { 1 }$, construct a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. Color the shape created in the same way as for figure $R _ { 1 }$ to obtain figure $R _ { 2 }$.
Continue this process. Let $S _ { n }$ be the area of the colored region in figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$

In the following graph, how many 1's are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 10
(2) 14
(3) 18
(4) 22
(5) 26

As shown in the figure, for a square ABCD with side length 5, let the five division points of diagonal BD be $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ in order from point B. Draw squares with diagonals $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ and circles with diameters $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$, then color the figure-eight-shaped region to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ closest to point A, and $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, and in these 2 new squares, draw figure-eight-shaped figures using the same method as for $R _ { 1 }$ and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped figures using the same method as obtaining $R _ { 2 }$ from $R _ { 1 }$ 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 figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [3 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, 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 circle $O$ with diameter AB of length 4. Let C be the center of the circle, and let D and P be the midpoints of segments AC and BC, respectively. Let E and Q be the points where the perpendicular bisector of segment AC and the perpendicular bisector of segment BC meet the upper semicircle of circle $O$, respectively. Draw a square DEFG with side DE that meets circle $O$ at point A and has diagonal DF, and draw a square PQRS with side PQ that meets circle $O$ at point B and has diagonal PR. Color the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square DEFG, and the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square PQRS to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, draw circle $O _ { 1 }$ centered at point F with radius $\frac { 1 } { 2 } \overline { \mathrm { DE } }$, and circle $O _ { 2 }$ centered at point R with radius $\frac { 1 } { 2 } \overline { \mathrm { PQ } }$. Color 2 $\square$-shaped figures and 2 $\square$-shaped figures created in the same way as obtaining figure $R _ { 1 }$ on the two circles $O _ { 1 }$ and $O _ { 2 }$ to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained the $n$-th time. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 12 \pi - 9 \sqrt { 3 } } { 10 }$
(2) $\frac { 8 \pi - 6 \sqrt { 3 } } { 5 }$
(3) $\frac { 32 \pi - 24 \sqrt { 3 } } { 15 }$
(4) $\frac { 28 \pi - 21 \sqrt { 3 } } { 10 }$
(5) $\frac { 16 \pi - 12 \sqrt { 3 } } { 5 }$

What is the value of $\lim _ { n \rightarrow \infty } \frac { \frac { 5 } { n } + \frac { 3 } { n ^ { 2 } } } { \frac { 1 } { n } - \frac { 2 } { n ^ { 3 } } }$?

As shown in the figure, there is a semicircle with diameter AB of length 2. Two points $\mathrm { P } , \mathrm { Q }$ are taken on arc AB such that $\angle \mathrm { PAB } = \theta , \angle \mathrm { QBA } = 2 \theta$, and the intersection of two line segments $\mathrm { AP } , \mathrm { BQ }$ is denoted R. Points S on segment AB, point T on segment BR, and point U on segment AR are chosen such that segment UT is parallel to segment AB and triangle STU is equilateral. Let $f ( \theta )$ be the area of the region enclosed by two line segments $\mathrm { AR } , \mathrm { QR }$ and arc AQ, and let $g ( \theta )$ be the area of triangle STU. When $\lim _ { \theta \rightarrow 0 + } \frac { g ( \theta ) } { \theta \times f ( \theta ) } = \frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Given that $0 < \theta < \frac { \pi } { 6 }$ and $p$ and $q$ are coprime natural numbers.) [4 points]

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions. (가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$ (나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$. When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

For a polynomial function $f ( x )$, define the function $g ( x )$ as follows: $$g ( x ) = \begin{cases} x & ( x < - 1 \text{ or } x > 1 ) \\ f ( x ) & ( - 1 \leq x \leq 1 ) \end{cases}$$ For the function $h ( x ) = \lim _ { t \rightarrow 0 + } g ( x + t ) \times \lim _ { t \rightarrow 2 + } g ( x + t )$, which of the following statements in the given options are correct? [4 points]
ㄱ. $h ( 1 ) = 3$ ㄴ. The function $h ( x )$ is continuous on the set of all real numbers. ㄷ. If the function $g ( x )$ is decreasing on the closed interval $[ - 1, 1 ]$ and $g ( - 1 ) = - 2$, then the function $h ( x )$ has a minimum value on the set of all real numbers.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

There is one bag and two boxes A and B. The bag contains 4 cards with the numbers $1, 2, 3, 4$ written on them, one number per card. Box A contains more than 8 white balls and more than 8 black balls, and box B is empty. Using this bag and the two boxes A and B, the following trial is performed.
A card is randomly drawn from the bag, the number on the card is confirmed, and the card is returned to the bag.
If the confirmed number is 1, 1 white ball from box A is placed into box B. If the confirmed number is 2 or 3, 1 white ball and 1 black ball from box A are placed into box B. If the confirmed number is 4, 2 white balls and 1 black ball from box A are placed into box B.
After repeating this trial 4 times, given that the total number of balls in box B is 8, find the probability that the number of black balls in box B is 2. [4 points]
(1) $\frac{3}{70}$
(2) $\frac{2}{35}$
(3) $\frac{1}{14}$
(4) $\frac{3}{35}$
(5) $\frac{1}{10}$

As shown in the figure, there is a right triangle ABC with $\overline { \mathrm { AB } } = 3$, $\overline { \mathrm { BC } } = 4$, and $\angle \mathrm { B } = \frac { \pi } { 2 }$. Let D be the point that divides segment AB internally in the ratio $2 : 1$, let E be the point where the circle centered at A with radius $\overline { \mathrm { AD } }$ meets segment AC, let F be the point where the line AB meets this circle other than D, and let G be a point on arc EF such that $\overline { \mathrm { CG } } = 2 \sqrt { 6 }$. When point H on the circle passing through the three points C, E, G satisfies $\angle \mathrm { HCG } = \angle \mathrm { BAC }$, what is the length of segment GH? [4 points]
(1) $\frac { 6 \sqrt { 15 } } { 5 }$
(2) $\frac { 38 \sqrt { 10 } } { 25 }$
(3) $\frac { 14 \sqrt { 3 } } { 5 }$
(4) $\frac { 32 \sqrt { 15 } } { 25 }$
(5) $\frac { 8 \sqrt { 10 } } { 5 }$

Solve for $x$:
$$\sin ( 2 x ) + \cos ^ { 2 } ( x ) = \frac { \ln ( x ) } { e ^ { 2 x } + 1 }$$

Solve for $x$ :
$$\sin ( 2 x ) + \cos ^ { 2 } ( x ) = \frac { \ln ( x ) } { e ^ { 2 x } + 1 }$$

Calculate the surface area of a torus with a major radius $R = 5$ units and a minor radius $r = 2$ units.

Calculate the surface area of a torus with a major radius $R = 5$ units and a minor radius $r = 2$ units.

Given a set of data with a multimodal distribution, calculate the skewness and kurtosis.

In a deck of cards, what is the probability that when drawing 10 cards, none of them are hearts?

Calculate the exact value of: $\cos \left( \frac { \pi } { 11 } \right)$

Translate the statement "For every real number $x$, there exists a real number $y$ such that $x ^ { 3 } + 3 y - 2 = 0$.

Translate the statement "For every real number $x$, there exists a real number $y$ such that $x ^ { 3 } + 3 y - 2 = 0$.

A particle in three-dimensional space moves along the path: $\vec { r } ( t ) = \left\langle t ^ { 2 } , \sin ( t ) , e ^ { t } \right\rangle$. Find the time when the particle changes direction.