System of inequalities $$\left\{ \begin{array} { l } \frac { x + 2 } { x ^ { 2 } - 4 x + 3 } \geqq 0 \\ \frac { 9 } { x - 8 } \leqq - 1 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

For two natural numbers $a , b ( a < b )$, the fractional inequality
$$\frac { x } { x - a } + \frac { x } { x - b } \leqq 0$$
is satisfied by exactly 2 integers $x$. What is the number of ordered pairs $( a , b )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

System of inequalities $$\left\{ \begin{array} { l } x ( x - 4 ) ( x - 5 ) \geqq 0 \\ \frac { x - 3 } { x ^ { 2 } - 3 x + 2 } \leqq 0 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Two quadratic expressions $f ( x ) , g ( x )$ with leading coefficient 1 have greatest common divisor $x + 3$ and least common multiple $x ( x + 3 ) ( x - 4 )$. How many integers $x$ satisfy the fractional inequality $\frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) } \leqq 0$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the fractional inequality in $x$ $$1 + \frac { k } { x - k } \leqq \frac { 1 } { x - 1 }$$ Find the value of the natural number $k$ such that the number of integers $x$ satisfying the inequality is 3. [3 points]

For the logarithmic inequality in $x$ $$\log _ { 5 } ( x - 1 ) \leq \log _ { 5 } \left( \frac { 1 } { 2 } x + k \right)$$ when the number of all integers $x$ satisfying this inequality is 3, what is the value of the natural number $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The number of elements in the set $$\left\{x : 0 \leqslant x \leqslant 2,\, \left|x - x^5\right| = \left|x^5 - x^6\right|\right\}$$ is
(A) 2
(B) 3
(C) 4
(D) 5

How many integers $x$ satisfy $2|x| + x < 10$?
(1) 13
(2) 14
(3) 15
(4) 16
(5) Infinitely many

$$\frac { - 5 } { 4 } < x < \frac { 7 } { 3 }$$
What is the sum of the integers $x$ that satisfy this inequality?
A) - 2
B) - 1
C) 0
D) 1
E) 2

$$-2 < x < 4$$
Given that, what is the greatest integer value that the expression $1 - x$ can take?
A) $-3$
B) $-2$
C) $-1$
D) 2
E) 3

a, b are real numbers and
$$\begin{aligned} & 0 < a < 3 a ^ { 2 } \\ & b - 1 = 6 a \end{aligned}$$
Given this, what is the smallest integer value that b can take?
A) 3
B) 4
C) 5
D) 6
E) 7

$$\mathrm { A } = \left\{ \mathrm { n } ( - 1 ) ^ { \mathrm { n } } : \mathrm { n } = 1,2,3 , \ldots , \mathrm { k } \right\}$$
The difference between the largest and smallest elements of the set is 25. Accordingly, how many positive elements does set A have?
A) 4
B) 6
C) 8
D) 10
E) 12

Integers a and b satisfy the inequality
$$1 < a < b - a < 5$$
Accordingly, what is the sum of the values that b can take?
A) 11
B) 14
C) 15
D) 16
E) 18

For real numbers $x$ and $y$,
$$\begin{aligned} & 3 < x < 12 \\ & \frac { x } { y } = \frac { 3 } { 2 } \end{aligned}$$
Given this, what is the sum of the integer values that $y$ can take?
A) 18
B) 21
C) 25
D) 28
E) 32

$( x - 1 ) ^ { 2 } < | x - 1 | + 6$\ What is the sum of the integers $x$ that satisfy this inequality?\ A) 2\ B) 3\ C) 4\ D) 5\ E) 6

For real number $x$
$$- 3 < 2 x < 7$$
Accordingly, what is the sum of the integer values that the expression $5 - x$ can take?
A) 5 B) 10 C) 15 D) 20 E) 25

The appearance of an application used to adjust the sound level of a computer, consisting of 100 equal units with a speaker symbol at the bottom, is given below.
The sound level of the computer

• when set to at least 1 and at most 32 units, the symbol appears as I)
• when set to at least 33 and at most 65 units, the symbol appears as I\textbullet)
• when set to at least 66 and at most 100 units, the symbol appears as I\textbullet))

On this computer, which is initially at a certain sound level, if the sound level is increased by 17 units, the symbol appears as I(\textbullet)), and if the initial sound level is decreased by 18 units, the symbol appears as I).
Accordingly, what is the sum of the integer values that the initial sound level can take in units?
A) 95
B) 96
C) 97
D) 98

Let $a$ be an integer. There are exactly 4 integer values of $x$ satisfying
$$0 < \left| x^{2} - 2x + 2 \right| - x^{2} - x < a$$
What is the sum of the different integer values that $a$ can take?
A) 33 B) 36 C) 39 D) 42 E) 45