Let $a, b, c, d$ be real numbers. It is known that the augmented matrices of two systems of linear equations $\left\{\begin{array}{l} ax + by = 2 \\ cx + dy = 1 \end{array}\right.$ and $\left\{\begin{array}{l} ax + by = -1 \\ cx + dy = -1 \end{array}\right.$, after the same row operations, become $\left[\begin{array}{cc|c} 1 & -1 & 3 \\ 0 & 1 & 2 \end{array}\right]$ and $\left[\begin{array}{cc|c} 1 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$ respectively. Then the solution to the system of equations $\left\{\begin{array}{l} ax + by = 0 \\ cx + dy = 1 \end{array}\right.$ is $x = $ 9-1, 9-2, $y = $ 9-3.

$$\frac { 1 } { 2 } - 3 a = \frac { 1 } { 8 } + 3 b$$
Given this, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 3 } { 4 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 1 } { 8 }$
D) $\frac { 5 } { 8 }$
E) $\frac { 4 } { 9 }$

$$\begin{aligned} & x ^ { 3 } - 2 y = 7 \\ & x ^ { 4 } - 2 x y = 21 \end{aligned}$$
Given this, what is $\mathbf { x }$?
A) 3
B) 5
C) 7
D) 9
E) 11

Let $\mathrm { a } , \mathrm { b } , \mathrm { x }$ and y be positive numbers such that
$$\begin{aligned} & \frac { x } { a } \cdot \frac { b } { y } = 2 \\ & \frac { a ^ { 2 } } { x ^ { 2 } } + \frac { b ^ { 2 } } { y ^ { 2 } } = 20 \end{aligned}$$
Given this, which of the following is the value of x in terms of a?
A) $\frac { a } { 2 }$
B) $\frac { 3 a } { 4 }$
C) $\frac { 3 a } { 5 }$
D) $\frac { 4 a } { 5 }$
E) $\frac { 5 a } { 6 }$

The sum of a three-digit number $ABC$ and a two-digit number $AB$ is 392.
Accordingly, what is the sum $\mathrm { A } + \mathrm { B } + \mathrm { C }$?
A) 7
B) 9
C) 11
D) 15
E) 19

$\frac{a - 1}{b} = \frac{c}{a}$
$$\frac{a}{c - 2} = \frac{b + 3}{a - 1}$$
Given that, what is the value of the expression $3c - 2b$?
A) 8 B) 9 C) 6 D) 3 E) 4

For distinct numbers a and b
$$\frac{a^{2}}{b} - \frac{b^{2}}{a} = b - a$$
Given that, what is the value of the expression $\frac{a}{b} + \frac{b}{a}$?
A) $-2$ B) $-1$ C) 0 D) 1 E) 4

$$\frac { a - 1 } { a - 3 } = \frac { a - 5 } { a - 4 }$$
Given that, what is a?
A) $\frac { 8 } { 5 }$
B) $\frac { 13 } { 4 }$
C) $\frac { 9 } { 4 }$

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 2 } - 4 y = - 7 \\ & y ^ { 2 } - 2 x = 2 \end{aligned}$$
Given this, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

For real numbers $x , y$ and $z$
$$\begin{aligned} & x \cdot y = 14 \\ & x \cdot z = 20 \\ & 3x + 2y + z = 24 \end{aligned}$$
Given that, what is x?
A) $\frac { 8 } { 3 }$
B) $\frac { 14 } { 5 }$
C) 3

$$\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { 2 - \sqrt { 4 - x } }$$
What is the value of this limit?
A) 3
B) 9
C) 12
D) 15
E) 16

$\mathbf { a } , \mathbf { b } , \mathbf { c }$ are non-zero real numbers and $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { a b }$. Given this,
$$\frac { a b + a c + b c + c ^ { 2 } } { a b c }$$
Which of the following is this expression equal to?
A) $\frac { a + 1 } { a }$
B) $\frac { b + 1 } { b }$
C) $\frac { c + 1 } { c }$
D) $\frac { b } { a }$
E) $\frac { b } { c }$

If Ahmet's salary is increased by half of Deniz's salary, then the sum of their salaries becomes 2 times Ahmet's initial salary.
If Ahmet's salary is A TL and Deniz's salary is D TL, what is the relationship between $A$ and $D$?
A) $5A = 8D$
B) $5A = 6D$
C) $4A = 5D$
D) $3A = 4D$
E) $2A = 3D$

In a laboratory, the following are known about a drug experiment conducted on male and female guinea pig mice.

• Male mice were given 1 tablet of medicine every 12 hours, and female mice were given 1 tablet every 8 hours.
• Male mice were given 0.5 gram tablets, and female mice were given 1 gram tablets.
• A total of 85 grams of medicine was given to these mice in one day, in the form of 95 tablets.

Accordingly, how many mice were used in the experiment?
A) 20
B) 25
C) 30
D) 35
E) 40

Stationery materials are to be distributed to students in a class. Enough materials are brought to the class so that each of the 36 students receives one pencil, one pencil sharpener, and one eraser. However, on the distribution day, since some students are absent, each student present in the class is given 3 pencils, 2 pencil sharpeners, and 1 eraser.
If a total of 42 items of these materials remain after distribution, how many pencil sharpeners are left?
A) 10
B) 11
C) 12
D) 13
E) 14

Let $\mathbf { k }$ be a nonzero real number such that
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } = ( 6 k ) ^ { 2 } \\ & ( x - 2 k ) ^ { 2 } + y ^ { 2 } = ( 2 k \sqrt { 5 } ) ^ { 2 } \end{aligned}$$
Accordingly, which of the following is the equivalent of $x ^ { 2 } - y ^ { 2 }$ in terms of $k$?
A) $13 \mathrm { k } ^ { 2 }$
B) $14 \mathrm { k } ^ { 2 }$
C) $15 k ^ { 2 }$

Sets $A$, $B$, and $C$ are defined as $$\begin{aligned}& A = \{ ( x , x ) : x \in \mathbb { R } \} \\& B = \{ ( x , 3 - x ) : x \in \mathbb { R } \} \\& C = \{ ( x , x + 4 ) : x \in \mathbb { R } \}\end{aligned}$$ Given that $( p , q ) \in A \cap B$ and $( r , s ) \in B \cap C$, $$\frac { p - r } { q + s }$$ what is the value of this expression?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 1 } { 4 }$\ C) $\frac { 3 } { 4 }$\ D) $\frac { 4 } { 5 }$\ E) $\frac { 2 } { 5 }$

Boxes A, B, C, and D contain a certain number of balls. The number of balls in box A is:

• equal to 2 times the number of balls in box B,
• equal to 3 times the number of balls in box C,
• equal to 4 times the number of balls in box D.

If one of the boxes contains 8 balls, how many balls are there in total in these boxes?
A) 30
B) 36
C) 40
D) 44
E) 50

Districts A, B, and C and the roads between these districts are shown in the figure below.
The road distances of points D and E, which are on these roads, to some districts are given in the tables on the signs.
Accordingly, what is the difference between the road distance from district C to district B and the road distance from district C to district A in km?
A) 6
B) 8
C) 10
D) 12
E) 14

\c{C}\i{}nar has a total of 78 pens, some of which are blue. He distributed these pens among three pen holders as follows.

• The number of pens in the pen holders is directly proportional to 3, 4, and 6.
• The number of blue pens in each pen holder is equal to each other.
• In one of the pen holders, the ratio of the number of blue pens to the total number of pens in that holder is $\frac{1}{2}$; in another pen holder, this ratio is $\frac{1}{3}$.

Accordingly, how many blue pens does \c{C}\i{}nar have in total?
A) 18
B) 24
C) 27
D) 30
E) 36

A group of students, each 7 years old, visited a botanical garden in 2015; another group of students, each 10 years old, visited in 2020. The official who guided the groups through the garden said about the same historical tree in the garden to both groups: ``The age of this tree is equal to the sum of all of your ages.''
From these two groups, if the number of students in the first group is 10 more than the number of students in the second group, how old is this tree in 2020?
A) 220
B) 230
C) 240
D) 250
E) 260

A rectangular towel has one side blue and the other side white. This towel is hung on a straight rack such that the short sides of the towel are parallel to the rack. The length of the non-overlapping part of the towel's sides is 6 cm when the towel is hung as in Figure 1; and 12 cm when hung as in Figure 2.
The ratio of the area of the blue side of the towel visible in Figure 1 to the area visible in Figure 2 is $\frac{5}{4}$.
Accordingly, what is the length of the long side of the towel in cm?
A) 24
B) 28
C) 30
D) 36
E) 40

The front view of a cabinet consisting of five compartments shown in the figure is square-shaped. The door of each compartment is a rectangle with equal areas.
One compartment has been filled with only shirts as shown in the figure.
Accordingly, how many times is the long side of the lid of the compartment containing the shirts compared to its short side?
A) $\frac{4}{3}$
B) $\frac{5}{3}$
C) $\frac{7}{4}$

A wooden piece in the shape of a square right prism with a square base has a base edge length equal to 2 times its height. When a cube with an edge length equal to the height of the wooden piece is removed from inside this wooden piece, the surface area of the resulting shape in the final state is 8 square units more than the surface area of the wooden piece in the initial state.
Accordingly, what is the volume of the wooden piece in the initial state in cubic units?
A) 32
B) 80
C) 108
D) 144
E) 256

The volume of a rectangular prism is equal to the product of its base area and height.
A closed glass container in the shape of a rectangular prism contains 360 cubic units of water. When the container is placed on a flat surface with different faces completely touching the surface, the height of the water is 2 units, 4 units, and 5 units respectively.
Accordingly, what is the volume of the container in cubic units?
A) 540
B) 720
C) 840
D) 960
E) 4080