For real numbers $\mathbf { a }$ and $\mathbf { b }$
$$b ^ { 2 } < a \cdot b < b - a$$
Given that, which of the following orderings is correct?
A) $a < 0 < b$ B) $b < 0 < a$ C) $0 < a < b$ D) $\mathrm { b } < \mathrm { a } < 0$ E) $a < b < 0$

Let $a , b , c$ be real numbers and $a \cdot b \cdot c > 0$ such that
$$\begin{aligned} & a \cdot b = - 2 | a | \\ & \frac { b } { c } = 3 | b | \end{aligned}$$
Given that $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { 0 }$, what is a?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 7 } { 3 }$
E) $\frac { 8 } { 3 }$

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

For integers a and b
$$16 ^ { a } \cdot 9 ^ { a } = 6 ^ { b } \cdot 8 ^ { 2 }$$
Given this equality, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 6
B) 9
C) 12
D) 15
E) 20

$$\mathrm { A } = \left\{ \mathrm { n } ( - 1 ) ^ { \mathrm { n } } : \mathrm { n } = 1,2,3 , \ldots , \mathrm { k } \right\}$$
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

On the real number line, numbers whose distance to point 2 is less than half the distance to point $-4$ form the solution set of which of the following inequalities?
A) $| x - 2 | < | x + 4 |$
B) $| x + 2 | < | x - 4 |$
C) $| 2 x - 4 | < | x + 4 |$
D) $| 2 x - 4 | < | x - 4 |$
E) $| 2 x + 4 | < | x + 4 |$

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

For nonzero real numbers $x$ and $y$, given that $y < x$ and $x ^ { 2 } < y ^ { 2 }$,\ I. $x \cdot y > 0$\ II. $x + y < 0$\ III. $\frac { 1 } { x } - \frac { 1 } { y } > 0$\ Which of the following statements are always true?\ A) Only I\ B) Only II\ C) I and II\ D) I and III\ E) II and III

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

$\frac { 6 x + 1 } { ( x + 1 ) ^ { 2 } } > 1$\ Which of the following is the set of all real numbers that satisfy this inequality?\ A) $( - 1,4 )$\ B) $( - 1,6 )$\ C) $( 0,4 )$\ D) $( 0 , \infty )$\ E) $( 2 , \infty )$

Let $\mathrm { a }$, $\mathrm { b }$ and $c$ be non-zero real numbers,
$$\begin{aligned} & \mathrm { p } : \mathrm { a } + \mathrm { b } = 0 \\ & \mathrm { q } : \mathrm { a } + \mathrm { c } < 0 \\ & \mathrm { r } : \mathrm { c } < 0 \end{aligned}$$
the propositions are given.
$$( p \wedge q ) \Rightarrow r$$
Given that the proposition is false; what are the signs of $\mathbf { a }$, $\mathbf { b }$ and $\mathbf { c }$ respectively?
A) $+$, $-$, $+$ B) $+$, $-$, $-$ C) $-$, $-$, $+$ D) $+$, $+$, $-$

In the Cartesian coordinate plane, the graphs of functions $f$, $g$ and $h$ whose domains consist of real numbers are given in the figure.
Accordingly, for $x \in [ - 2,2 ]$,
$$\begin{aligned} & f ( x ) \cdot g ( x ) > 0 \\ & g ( x ) \cdot h ( x ) < 0 \end{aligned}$$
the solution set of the system of inequalities is which of the following?
A) $( - 2 , - 1 )$ B) $( - 1,0 )$ C) $( 1,2 )$ D) $( - 2 , - 1 ) \cup ( 1,2 )$

A weather forecaster made the following statement during a live broadcast on Sunday evening.
"In our city where the temperature has been 5 degrees throughout this week, the weather will suddenly warm up starting tomorrow and winter will give way to spring-like weather. On Monday afternoon, the air temperature throughout the city will have increased by 6 to 10 degrees compared to the previous day."
Based on this information, which of the following inequalities expresses the range of values that the temperature in the city could take on Monday afternoon?
A) $|x - 13| \leq 2$
B) $|x - 10| \leq 6$
C) $|x - 6| \leq 5$
D) $|x - 1| \leq 6$
E) $|x - 11| \leq 2$

A positive number A is shown on the number line as in the figure.
Then, on this number line; numbers whose distance from 0 is equal to half the distance of number A from 0 are marked.
If the distance from one of the marked numbers to number A is 6 units, what is the sum of the possible values of number A?
A) 15
B) 16
C) 18
D) 20
E) 21

Let a be a real number. Regarding the inequality $x + 1 \leq a$, the following are known.

• $\mathrm { x } = 0$ satisfies this inequality.
• $x = 4$ does not satisfy this inequality.

Accordingly, what is the widest interval expressing the values that the number a can take?
A) $( 0,4 ]$
B) $[ 0,4 )$
C) $[ 1,4 ]$
D) $( 1,5 ]$
E) $[ 1,5 )$

For distinct real numbers $a , b$ and $c$,
$$\begin{aligned} & a + b = | a | \\ & b + c = | b | \end{aligned}$$
equalities are given. Accordingly; what is the correct ordering of the numbers $\mathbf { a , b }$ and c?
A) a < b < c
B) a $<$ c $<$ b
C) b $<$ a $<$ c
D) b $<$ c $<$ a
E) c $<$ a $<$ b

For real numbers $a$, $b$, and $c$
$$a - b < 0 < c < c - b$$
the inequality is given.
Accordingly, I. $a \cdot b \cdot c > 0$ II. $( a + c ) \cdot b > 0$ III. $b - a + c > 0$ which of these statements are always true?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III

The difference between the heights of a building and a tree on flat ground is 8 meters. After some time, the tree's height doubled and this difference became 3 meters.
Accordingly, the building's height I. 13 meters II. 16 meters III. 19 meters which of these values could it be?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

On the number line given below, the distance of K to 1 is equal to the distance of L to 2.
Accordingly, which of the following could be the value of the product $K \cdot L$?
A) A
B) B
C) C
D) D
E) E

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 $a$, $b$, $c$, and $d$ be real numbers such that
$$\begin{aligned} & a x ^ { 2 } + b x + 12 \geq 0 \\ & c x ^ { 2 } + d x + 24 \leq 0 \end{aligned}$$
To find the solution set of this system of inequalities, the following table is constructed and the solution set is found to be $[ - 2 , - 1 ] \cup [ 4,6 ]$.
What is the sum $a + b + c + d$?
A) 15
B) 16
C) 17
D) 18
E) 19

Bilge will choose two of the soup, salad, and fruit options given as one portion each at lunch based on the required calorie amount. Regarding the choices she can make, Bilge has calculated that the required calorie amount is
- exceeded when she chooses soup and fruit, - not exceeded when she chooses fruit and salad, - exactly met when she chooses salad and soup.
If the calories of one portion of soup, fruit, and salad are Ç, M, and S respectively, which of the following is the correct ordering of these values?
A) Ç $<$ M $\leq$ S B) Ç $\leq$ S $<$ M C) S $\leq$ Ç $<$ M D) S $<$ M $\leq$ Ç E) M $\leq$ S $<$ Ç

An instructor at a parachute jumping course gives the following explanation to the trainees:
"When jumping from an airplane at a height of 800 meters from the ground, you need to open your parachute 400 to 500 meters after jumping from the airplane in order to land safely on the ground."
Accordingly, which of the following inequalities expresses the values that the height from the ground when the parachute opens can take in order to land safely?
A) $|x - 350| \leq 50$ B) $|x - 300| \leq 100$ C) $|x - 250| \leq 150$ D) $|x - 200| \leq 200$ E) $|x - 150| \leq 250$

Let $x$ and $y$ be real numbers,
$$x^{2} \cdot y^{2} < x \cdot y < x^{2} \cdot y$$
Given this inequality.
Accordingly,
I. $x < 1$ II. $y < 1$ III. $x \cdot y < 1$
Which of these statements are true?
A) Only I B) Only II C) I and III D) II and III E) I, II and III