The figure on the right shows 6 semicircles with center at $( 1,1 )$ and radii of lengths $\frac { 1 } { 3 } , \frac { 2 } { 3 } , 1 , \frac { 4 } { 3 } , \frac { 5 } { 3 } , 2$ respectively. Three functions $$\begin{aligned} & y = \log _ { \frac { 1 } { 4 } } x \\ & y = \left( \frac { 2 } { 3 } \right) ^ { x } \\ & y = 3 ^ { x } \end{aligned}$$ Let $a , b , c$ be the number of intersection points where the graphs of these functions meet the semicircles, respectively. Which of the following correctly represents the relationship between $a , b , c$? (Here, $x \geqq 1$ and the semicircles include the endpoints of the diameter.) [4 points]
(1) $a < b < c$
(2) $a < c < b$
(3) $b < c < a$
(4) $c < a < b$
(5) $c < b < a$

For a positive number $a$, on the closed interval $[ - a , a ]$, the function
$$f ( x ) = \frac { x - 5 } { ( x - 5 ) ^ { 2 } + 36 }$$
has maximum value $M$ and minimum value $m$. Find the minimum value of $a$ such that $M + m = 0$. [4 points]

In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Function $$f ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } + x - 12 } { x - 3 } & ( x \neq 3 ) \\ a & ( x = 3 ) \end{array} \right.$$ When this function is continuous for all real numbers $x$, what is the value of $a$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

The figure on the right shows a circle with center at the origin O and radius 1, and the graph of a quadratic function $y = f ( x )$ passing through the point $( 0 , - 1 )$ on the coordinate plane. The equation $$\frac { 1 } { f ( x ) + 1 } - \frac { 1 } { f ( x ) - 1 } = \frac { 2 } { x ^ { 2 } }$$ has how many distinct real roots $x$? [3 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

For the function $f ( x ) = x ^ { 2 } - 4 x + a$ and the function $g ( x ) = \lim _ { n \rightarrow \infty } \frac { 2 | x - b | ^ { n } + 1 } { | x - b | ^ { n } + 1 }$, let $h ( x ) = f ( x ) g ( x )$. What is the value of $a + b$, the sum of the two constants $a , b$ such that the function $h ( x )$ is continuous for all real numbers $x$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

As shown in the figure, the graph of the cubic function $y = f ( x )$ is tangent to the $x$-axis at point $\mathrm { P } ( 2,0 )$ and meets the graph of the linear function $y = g ( x )$ only at point P. When $1 < f ( 0 ) < g ( 0 )$, what is the number of real roots of the equation $$f ( x ) + g ( x ) = \frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) }$$ ? [4 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

For the function $$f ( x ) = \begin{cases} x + 2 & ( x < - 1 ) \\ 0 & ( x = - 1 ) \\ x ^ { 2 } & ( - 1 < x < 1 ) \\ x - 2 & ( x \geqq 1 ) \end{cases}$$ which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [3 points]
$\langle$Remarks$\rangle$ ㄱ. $\lim _ { x \rightarrow 1 + 0 } \{ f ( x ) + f ( - x ) \} = 0$ ㄴ. The function $f ( x ) - | f ( x ) |$ is discontinuous at 1 point. ㄷ. There is no constant $a$ such that the function $f ( x ) f ( x - a )$ is continuous on the entire set of real numbers.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The graphs of the quadratic function $y = f ( x )$ and the cubic function $y = g ( x )$ are shown in the figure.
$f ( - 1 ) = f ( 3 ) = 0$, and the function $g ( x )$ has a local minimum value of $- 2$ at $x = 3$. What is the number of distinct real roots of the equation $\frac { g ( x ) + 2 } { f ( x ) } - \frac { 2 } { g ( x ) } = 1$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]
(1) $- 3$
(2) $- \frac { 3 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 6

A cubic function $f ( x )$ with leading coefficient 1 satisfies $f ( - x ) = - f ( x )$ for all real numbers $x$. When the equation $| f ( x ) | = 2$ has exactly 4 distinct real roots, what is the value of $f ( 3 )$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

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

The graph of a function $y = f ( x )$ defined on all real numbers is as shown in the figure, and a cubic function $g ( x )$ has leading coefficient 1 and $g ( 0 ) = 3$. When the composite function $( g \circ f ) ( x )$ is continuous on all real numbers, what is the value of $g ( 3 )$? [4 points]
(1) 31
(2) 30
(3) 29
(4) 28
(5) 27

For the two functions $$f(x) = \begin{cases} -1 & (|x| \geq 1) \\ 1 & (|x| < 1) \end{cases}, \quad g(x) = \begin{cases} 1 & (|x| \geq 1) \\ -x & (|x| < 1) \end{cases}$$ which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
Remarks ᄀ. $\lim_{x \rightarrow 1} f(x)g(x) = -1$ ㄴ. The function $g(x+1)$ is continuous at $x = 0$. ㄷ. The function $f(x)g(x+1)$ is continuous at $x = -1$.
(1) ᄀ
(2) ㄴ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ

As shown in the figure, the graphs of function $f ( x )$ defined on the closed interval $[ - 4,4 ]$ and function $g ( x ) = - \frac { 1 } { 2 } x + 1$ meet at three points, and the $x$-coordinates of these three points are $\alpha , \beta , 2$. The inequality $$\frac { g ( x ) } { f ( x ) } \leq 1$$ is satisfied. How many integers $x$ satisfy this inequality? (Here, $- 4 < \alpha < - 3,0 < \beta < 1$) [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

For the function
$$f ( x ) = \begin{cases} x + 1 & ( x \leq 0 ) \\ - \frac { 1 } { 2 } x + 7 & ( x > 0 ) \end{cases}$$
Find the sum of all real values of $a$ such that the function $f ( x ) f ( x - a )$ is continuous at $x = a$. [4 points]

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

Find the value of $\lim _ { x \rightarrow 0 } \frac { x ( x + 7 ) } { x }$. [3 points]

For the function $$f ( x ) = \begin{cases} 2 x + 10 & ( x < 1 ) \\ x + a & ( x \geq 1 ) \end{cases}$$ find the value of the constant $a$ such that $f$ is continuous on the entire set of real numbers. [3 points]

In the coordinate plane, let $f ( n )$ denote the number of triangles OAB satisfying the following conditions for a natural number $n$. Find the value of $f ( 1 ) + f ( 2 ) + f ( 3 )$. (Here, O is the origin.) [4 points] (가) The coordinates of point A are $\left( - 2,3 ^ { n } \right)$. (나) If the coordinates of point B are $( a , b )$, then $a$ and $b$ are natural numbers and satisfy $b \leq \log _ { 2 } a$. (다) The area of triangle OAB is at most 50.

What is the value of $\lim _ { x \rightarrow - 2 } \frac { ( x + 2 ) \left( x ^ { 2 } + 5 \right) } { x + 2 }$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

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

Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ What is the number of distinct real roots of the irrational equation $\sqrt { 4 - f ( x ) } = 1 - x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Two functions $$f ( x ) = \left\{ \begin{array} { l l } x + 3 & ( x \leq a ) \\ x ^ { 2 } - x & ( x > a ) \end{array} , \quad g ( x ) = x - ( 2 a + 7 ) \right.$$ Find the product of all real values of $a$ such that the function $f ( x ) g ( x )$ is continuous on the entire set of real numbers. [4 points]