Let the set $U$ be
$$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text { are positive numbers other than } 1 \right\}$$
Let the subset $S$ of $U$ be
$$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $A = \left( \begin{array} { l l } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two real numbers $a , b$ with $1 < a < b$,
$$\frac { 3 a } { \log _ { a } b } = \frac { b } { 2 \log _ { b } a } = \frac { 3 a + b } { 3 }$$
holds. Find the value of $10 \log _ { a } b$. [3 points]

For a natural number $n$, let $f ( n )$ be the mantissa of $\log n$. What is the number of elements in the set
$$A = \{ f ( n ) \mid 1 \leqq n \leqq 150 , n \text { is a natural number } \}$$
? [3 points]
(1) 131
(2) 133
(3) 135
(4) 137
(5) 139

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

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 filtered in 1 hour by shellfish A and B (in L) are $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relational equations 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

For a natural number $n ( n \geqq 2 )$, let the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet be $a _ { n }$ and $b _ { n }$ respectively ($a _ { n } < b _ { n }$). Which of the following statements in are correct? [4 points]
ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a natural number $n ( n \geqq 2 )$, let $a _ { n }$ and $b _ { n } \left( a _ { n } < b _ { n } \right)$ be the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$
(1) ᄀ
(2) ᄂ
(3) ᄃ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

For a natural number $n$ less than 10, when $\left( \frac { n } { 10 } \right) ^ { 10 }$ has a non-zero digit appearing for the first time in the sixth decimal place, what is the value of $n$? (Use $\log 2 = 0.3010 , \log 3 = 0.4771$ for calculations.) [4 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

Find the number of natural numbers $x$ that satisfy the logarithmic inequality $$\log _ { 2 } x \leqq \log _ { 4 } ( 12 x + 28 )$$ [3 points]

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

To determine the relative density of soil, a test device is inserted into the soil for investigation. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (Here, the units of $S$ and $R$ are metric ton/$\mathrm{m}^{2}$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

To determine the relative density of soil, a method of inserting a test device into the soil for investigation is used. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (where the units of $S$ and $R$ are metric ton $/ \mathrm { m } ^ { 2 }$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

When $\alpha$ is the root of the logarithmic equation $\log _ { 3 } ( x - 4 ) = \log _ { 9 } ( 5 x + 4 )$, find the value of $\alpha$. [3 points]

For a natural number $A$, let the characteristic of $\log A$ be $n$ and the mantissa be $\alpha$. Find the number of values of $A$ such that $n \leqq 2\alpha$ holds. (Given: $3.1 < \sqrt { 10 } < 3.2$) [4 points]

The sequence $\{a_n\}$ satisfies the following for all natural numbers $n$: $$\sum_{k=1}^{n} a_k = \log \frac{(n+1)(n+2)}{2}$$ Let $\sum_{k=1}^{20} a_{2k} = p$. Find the value of $10^p$. [4 points]

The female silkworm moth secretes pheromone to attract males. When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance of 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance of $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

The female silkworm moth secretes pheromone to attract males.
When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

For a positive number $x$, let the characteristic and mantissa of $\log x$ be $f ( x )$ and $g ( x )$, respectively. The number of natural numbers $n$ satisfying the two inequalities
$$f ( n ) \leq f ( 54 ) , \quad g ( n ) \leq g ( 54 )$$
is? [4 points]
(1) 42
(2) 44
(3) 46
(4) 48
(5) 50

Find the value of $x$ that satisfies the equation $\log _ { 3 } ( x - 11 ) = 3 \log _ { 3 } 2$. [3 points]

What is the value of $\log _ { 2 } 40 - \log _ { 2 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The temperature of a fire room changes over time. For a certain fire room, let the initial temperature be $T _ { 0 } \left( {}^{\circ}\mathrm{C} \right)$ and the temperature $t$ minutes after the fire starts be $T \left( {}^{\circ}\mathrm{C} \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8t + 1 ) \quad ($ where $k$ is a constant.$)$ In this fire room with an initial temperature of $20^{\circ}\mathrm{C}$, the temperature was $365^{\circ}\mathrm{C}$ after $\frac{9}{8}$ minutes from the start of the fire, and the temperature was $710^{\circ}\mathrm{C}$ after $a$ minutes from the start of the fire. What is the value of $a$? [3 points]
(1) $\frac{99}{8}$
(2) $\frac{109}{8}$
(3) $\frac{119}{8}$
(4) $\frac{129}{8}$
(5) $\frac{139}{8}$

The temperature of a fire room changes over time. Let the initial temperature of a certain fire room be $T _ { 0 } \left( { } ^ { \circ } \mathrm { C } \right)$, and the temperature $t$ minutes after the fire starts be $T \left( { } ^ { \circ } \mathrm { C } \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8 t + 1 ) \quad ($ where $k$ is a constant. $)$ In this fire room with an initial temperature of $20 ^ { \circ } \mathrm { C }$, the temperature was $365 ^ { \circ } \mathrm { C }$ at $\frac { 9 } { 8 }$ minutes after the fire started, and the temperature was $710 ^ { \circ } \mathrm { C }$ at $a$ minutes after the fire started. What is the value of $a$? [3 points]
(1) $\frac { 99 } { 8 }$
(2) $\frac { 109 } { 8 }$
(3) $\frac { 119 } { 8 }$
(4) $\frac { 129 } { 8 }$
(5) $\frac { 139 } { 8 }$

On the coordinate plane, for natural numbers $n$, consider the region $$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$ Let $a_n$ be the number of points in this region satisfying the following conditions. (가) The $x$-coordinate and $y$-coordinate are equal. (나) Both the $x$-coordinate and $y$-coordinate are integers. For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points]

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]