For a natural number $n$, define the function $f ( x )$ as $$f ( x ) = \begin{cases} \left| 3 ^ { x + 2 } - n \right| & ( x < 0 ) \\ \left| \log _ { 2 } ( x + 4 ) - n \right| & ( x \geq 0 ) \end{cases}$$ Let $g ( t )$ be the number of distinct real roots of the equation $f ( x ) = t$ for a real number $t$. Find the sum of all natural numbers $n$ such that the maximum value of the function $g ( t )$ is 4. [4 points]

For two natural numbers $a$ and $b$, the function $f(x)$ is defined as $$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$ For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$. $$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$ If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points]
(1) 51
(2) 52
(3) 53
(4) 54
(5) 55

Let $f ( x )$ be a cubic function with positive leading coefficient, and for a real number $t$, let the function $$g ( x ) = \left\{ \begin{array} { r r } - f ( x ) & ( x < t ) \\ f ( x ) & ( x \geq t ) \end{array} \right.$$ be continuous on the set of all real numbers and satisfy the following conditions. (가) For all real numbers $a$, the value of $\lim _ { x \rightarrow a + } \frac { g ( x ) } { x ( x - 2 ) }$ exists. (나) The set of natural numbers $m$ such that $\lim _ { x \rightarrow m + } \frac { g ( x ) } { x ( x - 2 ) }$ is negative is $\left\{ g ( - 1 ) , - \frac { 7 } { 2 } g ( 1 ) \right\}$. Find the value of $g ( - 5 )$. (Given that $g ( - 1 ) \neq - \frac { 7 } { 2 } g ( 1 )$) [4 points]

5. An odd function $f ( x )$ has domain $[ - 5,5 ]$. When $x \in [ 0,5 ]$, the graph of $f ( x )$ is shown in the figure on the right. The solution set of the inequality $f ( x ) < 0$ is $\_\_\_\_$.

7. As shown in the figure, the graph of function $f ( x )$ is the broken line $A C B$. The solution set of the inequality $f ( x ) \geqslant \log _ { 2 } ( x + 1 )$ is [Figure]
A. $\{ x \mid - 1 < x \leqslant 0 \}$
B. $\{ x \mid - 1 \leqslant x \leqslant 1 \}$
C. $\{ x \mid - 1 < x \leqslant 1 \}$
D. $\{ x \mid - 1 < x \leqslant 2 \}$

8. Given the function $f ( x ) = \begin{cases} 2 - | x | , & x \leq 2 \\ ( x - 2 ) ^ { 2 } , & x > 2 \end{cases}$, and function $g ( x ) = 3 - f ( 2 - x )$, then the number of intersections of the graphs of $y = f(x)$ and $y = g(x)$ is
(A) 2
(B) 3
(C) 4
(D) 5

II. Fill-in-the-Blank Questions: This section has 6 questions, each worth 5 points, for a total of 30 points.

Given the function $F(x) = \left\{\begin{array}{l}2 - |x|, \quad x \leq 2 \\ (x - 2)^2, \quad x > 2\end{array}\right.$ and function $g(x) = b - f(2 - x)$, where $b \in \mathbb{R}$. If the function $y = f(x) - g(x)$ has exactly 4 zeros, then the range of $b$ is
(A) $\left(\frac{7}{4}, +\infty\right)$
(B) $\left(-\infty, \frac{7}{4}\right)$
(C) $\left(0, \frac{7}{4}\right)$
(D) $\left(\frac{7}{4}, 2\right)$

10. The graph of the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ is shown in the figure. Then the correct conclusion is [Figure]
(A) $a > 0 , b < 0 , c > 0 , d > 0$
(B) $a > 0 , b < 0 , c < 0 , d > 0$
(C) $a < 0 , b < 0 , c < 0 , d > 0$
(D) $a > 0 , b > 0 , c > 0 , d < 0$

II. Fill in the Blank Questions

(11) $\lg \frac { 5 } { 2 } + 2 \lg 2 - \left( \frac { 1 } { 2 } \right) ^ { - 1 } =$ $\_\_\_\_$. (12) In $\triangle A B C$, $A B = \sqrt { 6 } , \angle A = 75 ^ { \circ } , \angle B = 45 ^ { \circ }$. Then $A C =$ $\_\_\_\_$. (13) In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { n } = a _ { n - 1 } + \frac { 1 } { 2 } ( n \geq 2 )$. Then the sum of the first 9 terms of the sequence $\left\{ a _ { n } \right\}$ equals $\_\_\_\_$. (14) In the rectangular coordinate system $x O y$, if the line $y = 2 a$ and the graph of the function $y = | x - a | - 1$ have only one intersection point, then the value of $a$ is $\_\_\_\_$. (15) $\triangle A B C$ is an equilateral triangle with side length 2. Given that vectors $\vec { a } , \vec { b }$ satisfy $\overrightarrow { A B } = 2 \vec { a } , \overrightarrow { A C } = 2 \vec { a } + \vec { b }$, then the correct conclusions among the following are $\_\_\_\_$. (Write out the serial numbers of all correct conclusions)
(1) $\vec { a }$ is a unit vector; (2) $\vec { b }$ is a unit vector; (3) $\vec { a } \perp \vec { b }$; (4) $\vec { b } \parallel \overrightarrow { B C }$; (5) $( 4 \vec { a } + \vec { b } ) \perp \overrightarrow { B C }$.

III. Solution Questions

As shown in the figure, rectangle $ABCD$ has sides $\mathrm { AB } = 2 , \mathrm { BC } = 1$, and $O$ is the midpoint of $AB$. Point $P$ moves along edges $\mathrm { BC } , \mathrm { CD }$, and $DA$, with $\angle \mathrm { BOP } = \mathrm { x }$. The sum of distances from moving point $P$ to points $A$ and $B$ is expressed as a function of $x$, denoted $f ( x )$. The graph of $f ( x )$ is approximately
(A), (B), (C), or (D) [as shown in figures]

10. Given the function $f ( x ) = \left\{ \begin{array} { l } x + \frac { 2 } { x } - 1 , x \geq 1 \\ \lg \left( x ^ { 2 } + 1 \right) , x < 1 \end{array} \right.$ , then $f ( f ( - 3 ) ) =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$ .

11. As shown in the figure, the rectangle's edge, is the midpoint, point moves along the edge, and moves with, denote, express the sum of distances from the moving point to two points as a function of, then the graph of is approximately
[Figure]
A.
[Figure]
B.
[Figure]
C.
[Figure]
D.

12. The number of zeros of the function $f(x) = 4\cos^2\frac{x}{2}\cos\left(\frac{\pi}{2} - x\right) - 2\sin x - |\ln(x+1)|$ is $\_\_\_\_$ .

12. Let $f ( x ) = \ln ( 1 + | x | ) - \frac { 1 } { 1 + x ^ { 2 } }$. The range of $x$ for which $f ( x ) > f ( 2 x - 1 )$ holds is
A. $\left( \frac { 1 } { 3 } , 1 \right)$
B. $\left( - \infty , \frac { 1 } { 3 } \right) \cup ( 1 , + \infty )$
C. $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
D. $\left( - \infty , - \frac { 1 } { 3 } \right) \cup \left( \frac { 1 } { 3 } , + \infty \right)$
II. Fill-in-the-Blank Questions: This section contains 4 questions, 5 points each, 20 points total

12. Given the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } , x \leq 1 \\ x + \frac { 6 } { x } - 6 , x > 1 \end{array} \right.$ , then $f [ f ( - 2 ) ] =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$.

13. The number of zeros of the function $f ( x ) = 2 \sin x \sin \left( x + \frac { \pi } { 2 } \right) - x ^ { 2 }$ is $\_\_\_\_$.

13. Given functions $f ( x ) = | \ln x | , g ( x ) = \left\{ \begin{array} { c } 0,0 < x \leq 1 \\ \left| x ^ { 2 } - 4 \right| - 2 , x > 1 \end{array} \right.$, then the number of real roots of the equation $| f ( x ) + g ( x ) | = 1$ is $\_\_\_\_$.

14. Let the function $f ( x ) = \left\{ \begin{array} { c c } 2 ^ { x } - a , & x < 1 , \\ 4 ( x - a ) ( x - 2 a ) , & x \geqslant 1 \text { .} \end{array} \right.$
(1) If $a = 1$, then the minimum value of $f ( x )$ is $\_\_\_\_$;
(2) If $f ( x )$ has exactly 2 zeros, then the range of the real number $a$ is $\_\_\_\_$.
III. Answer Questions (6 questions in total, 80 points. Solutions should include written explanations, calculation steps, or proof processes)

14. If the function $f ( x ) = \left| 2 ^ { x } - 2 \right| - b$ has two zeros, then the range of the real number $b$ is $\_\_\_\_$

14. Let F be a focus of the hyperbola $C: \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If there exists a point P on C such that the midpoint of segment PF is exactly an endpoint of its conjugate axis, then the eccentricity of C is $\_\_\_\_$

The graph of the function $y = 2 x ^ { 2 } - e ^ { | x | }$ on $[ - 2,2 ]$ is approximately
(A), (B), (C), (D) [as shown in the figures]

The range of the function $f(x) = \sqrt{x^2 - 2x - 3}$ is
A. $[-2, 2]$
B. $[-1, 1]$
C. $[0, 4]$
D. $[1, 3]$

The partial graph of the function $y = \frac{\sin 2x}{1 - \cos x}$ is approximately (see options A, B, C, D in the figures provided).

The graph of the function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately (see options A, B, C, D in the figures).

The graph of function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately
A. [Graph A]
B. [Graph B]
C. [Graph C]
D. [Graph D]

The graph of the function $y = - x ^ { 4 } + x ^ { 2 } + 2$ is approximately (See figures A, B, C, D in the original paper.)