The following explains the relationship between confidence interval, confidence level, and sample size.
There is a population following a normal distribution $N \left( m , \sigma ^ { 2 } \right)$. When a sample of size $n$ is randomly extracted from this population, the sample mean follows a normal distribution $\square$ (a). Using the distribution of this sample mean, let the confidence interval for the population mean $m$ with confidence level $\alpha$ be $a \leqq m \leqq b$. When the sample size is fixed at $n$ and the confidence level is set higher than $\alpha$, let the confidence interval be $c \leqq m \leqq d$. Then $d - c$ is $\square$ (b) than $b - a$. On the other hand, when the confidence level is fixed at $\alpha$ and the sample size is $2 n$, let the confidence interval be $e \leqq m \leqq f$. Then $f - e$ is $\square$ (c) times $b - a$.
What are the correct values for (a), (b), and (c) in the above process? [3 points]

	(a)	(b)
(1)	$N \left( m , \sigma ^ { 2 } \right)$	larger
(2)	$N \left( m , \sigma ^ { 2 } \right)$	smaller
(3)	$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$	smaller
(4)	$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$	larger
(5)	$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$	smaller

	(c)
(1)	$\frac{1}{2}$
(2)	$\frac{1}{2}$
(3)	$\frac{1}{\sqrt{2}}$
(4)	$\sqrt{2}$
(5)	$\frac{1}{\sqrt{2}}$

When sound passes through a building wall, a certain proportion is transmitted into the interior while the rest is reflected or absorbed. The ratio of sound transmitted into the interior is called the transmission rate. When the acoustic output of a speaker is $W$ (watts), the intensity $P$ (decibels) of sound transmitted into the interior at a distance of $r$ (m) from the speaker in a building with transmission rate $\alpha$ is as follows. $$\begin{aligned} & P = 10 \log \frac { \alpha W } { I _ { 0 } } - 20 \log r - 11 \\ & \text{(where } I _ { 0 } = 10 ^ { - 12 } \text{ (watts/m}^2\text{) and } r > 1 \text{.)} \end{aligned}$$ A speaker is emitting sound with an acoustic output of 100 (watts). When the intensity of sound transmitted into the interior of a building with transmission rate $\frac { 1 } { 100 }$ is 59 (decibels) or less, what is the minimum distance between the speaker and the building? (Assume that sound spreads uniformly in space and that factors other than transmission rate are not considered.) [4 points]
(1) $10 ^ { 2 } \mathrm{~m}$
(2) $10 ^ { \frac { 17 } { 8 } } \mathrm{~m}$
(3) $10 ^ { \frac { 13 } { 6 } } \mathrm{~m}$
(4) $10 ^ { \frac { 9 } { 4 } } \mathrm{~m}$
(5) $10 ^ { \frac { 5 } { 2 } } \mathrm{~m}$

A society where the proportion of the population aged 65 and over in the total population is 20\% or more is called a 'super-aged society'. In 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013$, $\log 1.04 = 0.0170$, $\log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010

How many additional edges must be drawn to make a regular 10-gon graph into a complete graph? [3 points]
(1) 25
(2) 35
(3) 45
(4) 55
(5) 65

How many different spanning trees does the graph on the right have? [3 points]
(1) 10
(2) 12
(3) 15
(4) 18
(5) 21

The following table shows the tasks, task times, and prerequisite tasks required to repair the interior of a moving house.

Task	Task Time (minutes)	Prerequisite Task
Wallpaper (A)	210	None
Light Bulb Replacement (B)	20	A
Bathroom Repair (C)	60	B
Kitchen Repair (D)	100	B
Curtain Replacement (E)	50	A
Flooring Replacement (F)	150	C, D

What is the minimum time required to complete all the tasks? [4 points]
(1) 260 minutes
(2) 370 minutes
(3) 440 minutes
(4) 480 minutes
(5) 530 minutes

For a function $y = f ( x )$ defined on all real numbers, let $N ( f )$ denote the smallest natural number $k$ such that the function $y = x ^ { k } f ( x )$ is continuous at $x = 0$. For example,
$$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{array} \text { then } N ( f ) = 2 \right. \text { . }$$
For the following functions $g _ { i } ( i = 1,2,3 )$, let $N \left( g _ { i } \right) = a _ { i }$. Which correctly represents the order of $a _ { i }$? [3 points]
$$\begin{aligned} & g _ { 1 } ( x ) = \begin{cases} \frac { | x | } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 2 } ( x ) = \begin{cases} - x ^ { 2 } + 1 & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 3 } ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \end{aligned}$$
(1) $a _ { 1 } = a _ { 2 } < a _ { 3 }$
(2) $a _ { 1 } < a _ { 2 } = a _ { 3 }$
(3) $a _ { 1 } = a _ { 2 } = a _ { 3 }$
(4) $a _ { 2 } = a _ { 3 } < a _ { 1 }$
(5) $a _ { 3 } < a _ { 1 } = a _ { 2 }$

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) $\frac { 2 } { 5 }$

Graph $G$ has vertices $1, 2, 3, 4, 5, 6, 7, 8$, and all edges connect two distinct vertices that have a divisor or multiple relationship. How many vertices in graph $G$ have degree 3? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $A$ be the adjacency matrix of a graph $G$ with 5 vertices. The following represents $A^2$. $$\left( \begin{array} { l l l l l } 4 & 3 & 3 & 2 & 2 \\ 3 & 4 & 3 & 2 & 2 \\ 3 & 3 & 4 & 2 & 2 \\ 2 & 2 & 2 & 3 & 3 \\ 2 & 2 & 2 & 3 & 3 \end{array} \right)$$ Choose all correct statements about graph $G$ from the given options. [4 points]
Options ㄱ. There are 2 vertices with degree 3. ㄴ. It has a Hamiltonian circuit. ㄷ. There are at least 2 paths consisting of 2 edges connecting any two distinct vertices.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The following shows a tree with 1023 vertices, where consecutive natural numbers from 1 to 1023 are assigned to each vertex according to a rule.
Let $M(a, b)$ be the maximum natural number corresponding to a vertex that is commonly included in both the path from the vertex corresponding to 1 to the vertex corresponding to $a$ and the path from the vertex corresponding to 1 to the vertex corresponding to $b$.
For example, $M(4, 11) = 2$ and $M(7, 12) = 3$.
If $M(33, 79) = k$, find the value of $10k$. [4 points]

A square is divided into three equal parts horizontally to create [Figure 1] and divided into three equal parts vertically to create [Figure 2]. [Figure 1] and [Figure 2] are alternately attached and continued to create a figure as shown below. As shown in the figure, let A be the top-left vertex of the first attached [Figure 1], and let $\mathrm { B } _ { n }$ be the bottom-right vertex of the figure created by attaching $n$ figures in total (combining the number of [Figure 1]s and [Figure 2]s). Let $a _ { n }$ be the number of shortest paths from vertex A to vertex $\mathrm { B } _ { n }$ along the lines. What is the value of $a _ { 3 } + a _ { 7 }$? [4 points]
(1) 26
(2) 28
(3) 30
(4) 32
(5) 34

As shown below, a rectangle with width 6 and height 8 has a circle drawn inside with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle, creating figure $R _ { 1 }$. In 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, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 2 }$. In each of the four congruent rectangles in figure $R _ { 2 }$, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the width and height 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) $\frac { 25 } { 9 } \pi$

There is a regular tetrahedron OABC with edge length 6. Let $S _ { 1 } , S _ { 2 } , S _ { 3 }$ be the orthogonal projections onto plane ABC of the three circles inscribed in triangles $\triangle \mathrm { OAB } , \triangle \mathrm { OBC } , \triangle \mathrm { OCA }$ respectively. As shown in the figure, let $S$ be the area of the dark region enclosed by the three figures $S _ { 1 } , S _ { 2 } , S _ { 3 }$. Find the value of $( S + \pi ) ^ { 2 }$. [4 points]

On plane $\alpha$, there is a right isosceles triangle ABC with $\angle \mathrm { A } = 90 ^ { \circ }$ and $\overline { \mathrm { BC } } = 6$. A point P outside plane $\alpha$ is at a distance of 4 from the plane, and the foot of the perpendicular from P to plane $\alpha$ is point A. What is the distance from point P to line BC? [3 points]
(1) $3 \sqrt { 2 }$
(2) 5
(3) $3 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$
(5) 6

Shellfish filter suspended matter. When the water temperature is $t \left( { } ^ { \circ } \mathrm { C } \right)$ and the individual weight is $w ( \mathrm {~g} )$, the amounts (in L) filtered in 1 hour by shellfish A and B are denoted as $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relationships hold. $$\begin{aligned} & Q _ { \mathrm { A } } = 0.01 t ^ { 1.25 } w ^ { 0.25 } \\ & Q _ { \mathrm { B } } = 0.05 t ^ { 0.75 } w ^ { 0.30 } \end{aligned}$$ When the water temperature is $20 ^ { \circ } \mathrm { C }$ and the individual weights of shellfish A and B are each 8 g, the value of $\frac { Q _ { \mathrm { A } } } { Q _ { \mathrm { B } } }$ is $2 ^ { a } \times 5 ^ { b }$. What is the value of $a + b$? (Here, $a$ and $b$ are rational numbers.) [3 points]
(1) 0.15
(2) 0.35
(3) 0.55
(4) 0.75
(5) 0.95

There is a computer game where two dolls A and B are dressed in shirts and pants with undetermined colors, and then the colors of the clothes are determined. There are 3 shirts and 3 pants of different shapes, and the color of each piece of clothing is determined to be either red or green. A shirt and pants put on one doll are not put on the other doll. The colors of doll A's shirt and pants are determined to be different, and the colors of doll B's shirt and pants are also determined to be different. In this game, when dressing dolls A and B in shirts and pants and determining their colors, what is the number of possible outcomes? [4 points]
(1) 252
(2) 216
(3) 180
(4) 144
(5) 108

As shown in the figure, a square A with side length 2 and a square B with side length 1 have sides parallel to each other, and the intersection point of the two diagonals of A coincides with the intersection point of the two diagonals of B. Let R be the region of A and its interior excluding the interior of B.
For a natural number $n \geqq 2$, small squares with side length $\frac { 1 } { n }$ are drawn in R according to the following rule. (가) One side of each small square is parallel to a side of A. (나) The interiors of the small squares do not overlap with each other.
According to such rules, let $a _ { n }$ be the maximum number of small squares with side length $\frac { 1 } { n }$ that can be drawn in R. For example, $a _ { 2 } = 12$ and $a _ { 3 } = 20$. When $\lim _ { n \rightarrow \infty } \frac { a _ { 2 n + 1 } - a _ { 2 n } } { a _ { 2 n } - a _ { 2 n - 1 } } = c$, find the value of $100 c$. [4 points]

As shown in the figure, there are two circular disks with distance between centers $\sqrt { 3 }$ and radius 1, and a plane $\alpha$. The line $l$ passing through the centers of each disk is perpendicular to the planes of the two disks and makes an angle of $60 ^ { \circ }$ with plane $\alpha$. When sunlight shines perpendicular to plane $\alpha$ as shown in the figure, what is the area of the shadow cast by the two disks on plane $\alpha$? (Note: the thickness of the disks is negligible.) [4 points]
(1) $\frac { \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 8 }$
(2) $\frac { 2 } { 3 } \pi + \frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 1 } { 8 }$
(4) $\frac { 4 } { 3 } \pi + \frac { \sqrt { 3 } } { 16 }$
(5) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 4 }$

For a natural number $m$, there are blocks in the shape of identical cubes stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following procedure is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove $\frac { 1 } { 2 }$ of the blocks in that column from the column.
After completing all block removal procedures, let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$. $$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$ Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]

Among the partitions of the natural number 7, how many distinct partitions can be expressed as the sum of natural numbers not exceeding 3? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

A newly constructed building has 6 offices A, B, C, D, E, F. The cost required to establish a computer network between offices is shown in the table. What is the minimum cost required to establish a computer network so that all 6 offices are connected through the network? [3 points] (Unit: 1 million won)

What is the sum of all entries of the matrix representing the connection relationships between vertices of the following graph? [3 points]
(1) 6
(2) 8
(3) 10
(4) 12
(5) 14

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) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2

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