For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$ respectively. Find the number of all ordered pairs $( a , b )$ satisfying the following conditions. For example, $a = 4 , b = 5$ satisfies the following conditions. [4 points] (가) $2 \leq a \leq 10, 2 \leq b \leq 10$ (나) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

In the coordinate plane, for a natural number $n$, let $a _ { n }$ be the number of points in the region
$$\left\{ ( x , y ) \mid 2 ^ { x } - n \leq y \leq \log _ { 2 } ( x + n ) \right\}$$
that satisfy the following conditions.
(a) The $x$-coordinate and $y$-coordinate are equal.
(b) The $x$-coordinate and $y$-coordinate are both integers. For example, $a _ { 1 } = 2$ and $a _ { 2 } = 4$. Find the value of $\sum _ { n = 1 } ^ { 30 } a _ { n }$. [4 points]

For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
What is the value of $\lim _ { n \rightarrow \infty } \frac { \left( \frac { a } { 3 } \right) ^ { n + k } } { \left( \frac { a } { 3 } \right) ^ { n + 1 } + 1 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a certain financial product, the expected asset $W$ after $t$ years of investing an initial asset $W _ { 0 }$ is given as follows: $$W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right)$$ (where $W _ { 0 } > 0 , t \geq 0$, and $a$ is a constant.) When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [4 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

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

Find the sum of all natural numbers $x$ satisfying the inequality $\left( \frac { 1 } { 2 } \right) ^ { x - 5 } \geq 4$. [3 points]

What is the maximum value of the function $f ( x ) = 1 + \left( \frac { 1 } { 3 } \right) ^ { x - 1 }$ on the closed interval $[ 1,3 ]$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) 2
(3) $\frac { 7 } { 3 }$
(4) $\frac { 8 } { 3 }$
(5) 3

When the graph of the function $y = 2 ^ { x } + 2$ is translated in the $x$-direction by $m$ units, and this graph is symmetric to the graph of the function $y = \log _ { 2 } 8 x$ translated in the $x$-direction by 2 units with respect to the line $y = x$, what is the value of the constant $m$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

When the graphs of the quadratic function $y = f ( x )$ and the linear function $y = g ( x )$ are as shown in the figure, the sum of all natural numbers $x$ satisfying the inequality $$\left( \frac { 1 } { 2 } \right) ^ { f ( x ) g ( x ) } \geq \left( \frac { 1 } { 8 } \right) ^ { g ( x ) }$$ is? [4 points] [Figure]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

What is the value of $\lim _ { x \rightarrow 0 } \frac { 6 x } { e ^ { 4 x } - e ^ { 2 x } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let A be the point where the graph of the exponential function $y = a ^ { x } ( a > 1 )$ meets the line $y = \sqrt { 3 }$. For the point $\mathrm { B } ( 4,0 )$, if the line OA and the line AB are perpendicular to each other, what is the product of all values of $a$? (Here, O is the origin.) [4 points]
(1) $3 ^ { \frac { 1 } { 3 } }$
(2) $3 ^ { \frac { 2 } { 3 } }$
(3) 3
(4) $3 ^ { \frac { 4 } { 3 } }$
(5) $3 ^ { \frac { 5 } { 3 } }$

For a constant $k$ with $k > 1$, there is a sequence $\left\{ a _ { n } \right\}$ satisfying the following conditions.
For all natural numbers $n$, $a _ { n } < a _ { n + 1 }$ and the slope of the line passing through two points $\mathrm { P } _ { n } \left( a _ { n } , 2 ^ { a _ { n } } \right)$ and $\mathrm { P } _ { n + 1 } \left( a _ { n + 1 } , 2 ^ { a _ { n + 1 } } \right)$ on the curve $y = 2 ^ { x }$ is $k \times 2 ^ { a _ { n } }$.
Let $\mathrm { Q } _ { n }$ be the point where the line passing through $\mathrm { P } _ { n }$ parallel to the $x$-axis and the line passing through $\mathrm { P } _ { n + 1 }$ parallel to the $y$-axis meet, and let $A _ { n }$ be the area of triangle $\mathrm { P } _ { n } \mathrm { Q } _ { n } \mathrm { P } _ { n + 1 }$. The following is the process of finding $A _ { n }$ when $a _ { 1 } = 1$ and $\frac { A _ { 3 } } { A _ { 1 } } = 16$.
Since the slope of the line passing through two points $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ is $k \times 2 ^ { a _ { n } }$, $$2 ^ { a _ { n + 1 } - a _ { n } } = k \left( a _ { n + 1 } - a _ { n } \right) + 1$$ Thus, for all natural numbers $n$, $a _ { n + 1 } - a _ { n }$ is a solution of the equation $2 ^ { x } = k x + 1$. Since $k > 1$, the equation $2 ^ { x } = k x + 1$ has exactly one positive real root $d$. Therefore, for all natural numbers $n$, $a _ { n + 1 } - a _ { n } = d$, and the sequence $\left\{ a _ { n } \right\}$ is an arithmetic sequence with common difference $d$. Since the coordinates of point $\mathrm { Q } _ { n }$ are $\left( a _ { n + 1 } , 2 ^ { a _ { n } } \right)$, $$A _ { n } = \frac { 1 } { 2 } \left( a _ { n + 1 } - a _ { n } \right) \left( 2 ^ { a _ { n + 1 } } - 2 ^ { a _ { n } } \right)$$ Since $\frac { A _ { 3 } } { A _ { 1 } } = 16$, the value of $d$ is (가), and the general term of the sequence $\left\{ a _ { n } \right\}$ is $$a _ { n } = \text { (나) }$$ Therefore, for all natural numbers $n$, $A _ { n } =$ (다).
When the number corresponding to (가) is $p$, and the expressions corresponding to (나) and (다) are $f ( n )$ and $g ( n )$ respectively, what is the value of $p + \frac { g ( 4 ) } { f ( 2 ) }$? [4 points]
(1) 118
(2) 121
(3) 124
(4) 127
(5) 130

Solve the equation $3^{x-8} = \left(\frac{1}{27}\right)^x$ for the real number $x$. [3 points]

Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions. For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$. What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]

For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points]
(1) $2 ^ { \frac { 13 } { 6 } }$
(2) $2 ^ { \frac { 7 } { 3 } }$
(3) $2 ^ { \frac { 5 } { 2 } }$
(4) $2 ^ { \frac { 8 } { 3 } }$
(5) $2 ^ { \frac { 17 } { 6 } }$

Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$. When $a \times b = \frac { q } { p }$, find the value of $p + q$. (Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]

15. Among the following functions, which one is both an even function and monotonically decreasing on the interval $(0, +\infty)$?
(A) $y = x^{-2}$
(B) $y = x^{-1}$
(C) $y = x^2$
(D) $y = x^{\frac{1}{3}}$

21. (Total: 14 points; Part 1: 6 points; Part 2: 8 points) Given the function $f(x) = a \cdot 2^x + b \cdot 3^x$, where constants $a, b$ satisfy $a \cdot b \neq 0$
(1) If $a \cdot b > 0$, determine the monotonicity of function $f(x)$;
(2) If $a \cdot b < 0$, find the range of $x$ when $f(x+1) > f(x)$.

7. The solution set of the inequality $2 ^ { x ^ { 2 } - x } < 4$ is $\_\_\_\_$ .

7. Given the function $f ( x ) = 2 ^ { | x - m | } - 1$ defined on $\mathbb{R}$ (where $m$ is a real number), let $a = f \left( \log _ { 0.5 } 3 \right)$, $b = f \left( \log _ { 2 } 5 \right)$, $c = f ( 2m )$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $c < a < b$
(C) $a < c < b$
(D)

Given the function $\mathrm{f}(x) = 2^{|x-1|} - 1$ defined on $\mathbb{R}$ (where m is a real number) is an even function, let $\mathrm{a} = \mathrm{f}(\log_{0.5}3)$, $b = f(\log_2 5)$, $c = f(2m)$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $c < a < b$
(D) $c < b < a$

9. Let $f ( x ) = \ln x , 0 < a < b$. If $p = f ( \sqrt { a b } ) , q = f \left( \frac { a + b } { 2 } \right)$, $r = \frac { 1 } { 2 } ( f ( a ) + f ( b ) )$, then the correct relation is
A. $q = r < p$
B. $q = r > p$
C. $p = r < q$
D. $p = r > q$

Among the three numbers $2 ^ { - 3 }$, $3 ^ { \frac { 1 } { 2 } }$, $\log _ { 2 } 5$, the largest is

13. Given that the graph of function $f ( x ) = a x ^ { 3 } - 2 x$ passes through the point $( - 1,4 )$, then $a = $ $\_\_\_\_$ .