$$\begin{aligned} & a = 5 ! \cdot 9 ! \\ & b = 6 ! \cdot 8 ! \\ & c = 7 ! \cdot 7 ! \end{aligned}$$
Given that, which of the following orderings is correct?
A) $a < b < c$ B) $a < c < b$ C) $b < c < a$ D) $c < a < b$ E) $c < b < a$

Below is a regular pentagon with vertex points A1, A2, A3, A4, and A5.
The $\otimes$ operation on the vertices of this pentagon is defined as
- for each vertex A: $\mathrm { A } \otimes \mathrm { A } = \mathrm { A }$ - for different vertices $A$ and $B$: $A \otimes B$ is the vertex point located on the perpendicular bisector of the line segment connecting points $A$ and $B$.
Given that $\left( A _ { 1 } \otimes A _ { 3 } \right) \otimes X = A _ { 5 }$, which of the following is vertex $X$?
A) $\mathrm { A } _ { 1 }$ B) $A_2$ C) $A_3$ D) $A_4$ E) $A_5$

$$\begin{array} { l l } p : & x = 0 \\ q : & y = 0 \end{array}$$
The following propositions are given.
Accordingly, for real numbers x and y
I. $x \cdot y = 0$ II. $x + y = 0$ III. $x ^ { 2 } + y ^ { 2 } = 0$
Which of the following propositions are equivalent to the proposition $\mathbf { p } \wedge \mathbf { q }$?
A) Only II B) Only III C) I and II D) I and III E) II and III

If a three-digit natural number with non-zero digits is divisible without remainder by each digit in each of its places, this number is called a "proper number."
If the number 3A4 is a proper number, what is the sum of the values that A can take?
A) 7 B) 8 C) 10 D) 13 E) 15

80\% of the students in a class can play guitar.
Given that 80\% of the students in the class are male, what is the minimum percentage of students who can play guitar that are male?
A) 64 B) 70 C) 72 D) 75 E) 80

Eight balls numbered 1 to 8 will be placed in two boxes with four balls in each box according to the following rules.
- The sum of the numbers of the balls in the boxes is equal to each other. - Each box contains one ball whose number is divisible by 3.
Accordingly, what is the product of the numbers of the balls in the box containing ball number 2?
A) 120 B) 192 C) 240 D) 360 E) 384

A vehicle starting with a full tank uses two-thirds of the gasoline in its tank when it stops at a gas station and adds half a tank of gasoline and continues its journey.
When the vehicle has traveled 900 km from the beginning, its gasoline runs out.
Given that the vehicle's gasoline consumption is constant throughout the journey, how many km is the distance the vehicle traveled between the starting point and the gas station?
A) 300 B) 360 C) 400 D) 450 E) 480

In a competition, a jury consisting of three people votes yes or no to the contestants. In this competition with 20 participants, a contestant must receive at least two yes votes to be successful.
In this competition where the jury members gave a total of 30 yes votes, 8 contestants were successful and no contestant received three no votes.
Accordingly, how many contestants received three yes votes?
A) 1 B) 2 C) 3 D) 4 E) 5

The weight of three of the balls $A, B, C$ and $D$ is the same. On a balance scale with equal arms:
- when balls $A$ and $B$ are placed on the left pan and balls $C$ and $D$ are placed on the right pan, the left pan is heavier, - when balls A and C are placed on the left pan and balls B and D are placed on the right pan, the left pan is again heavier.
Accordingly,
I. Balls A and B have equal weight. II. Balls B and C have equal weight. III. Ball A is heavier than ball D.
Which of the following statements are always true?
A) Only I B) Only II C) I and III D) II and III E) I, II and III

Passengers traveling on an airplane took at most one of the tea and coffee offered. From these passengers:
- the number of passengers who took tea is 3 times the number of passengers who took coffee, - the number of passengers who took neither of the tea and coffee refreshments is one-third of the total number of passengers.
Given that the number of passengers who did not take tea is 72, how many passengers did not take coffee?
A) 90 B) 96 C) 100 D) 108 E) 120

In an egg production farm, Ayhan and Burcu perform the work of arranging eggs in crates and packaging these crates.
- Ayhan arranges 3 crates per minute, while Burcu arranges 4 crates per minute. - Ayhan packages 6 crates per minute, while Burcu packages 5 crates per minute.
Ayhan arranged eggs in some crates and packaged these crates. During this time, Burcu arranged 60 crates of eggs and packaged these crates.
Accordingly, how many crates of eggs did Ayhan arrange?
A) 48 B) 50 C) 54 D) 60 E) 66

Alper will attend a meeting at his workplace that will be held at 08.00 in the morning. Leaving home one hour before the meeting time, Alper adjusts his walking speed so that he will arrive at the workplace in 1 hour.
When he reaches the midpoint of the road, Alper realizes he forgot his file at home, retrieves his file at a constant speed by running, and arrives at the workplace exactly on time by running at the same speed without stopping.
Given that Alper used the same route between home and workplace throughout his entire movement, at what time did he retrieve his file from home?
A) 07.36 B) 07.40 C) 07.42 D) 07.45 E) 07.48

Below is a moving device consisting of two small and large disks with the same centers and symbols placed at equal intervals on them. A rectangular fixed indicator is placed on top of this device.
These two disks moving at constant speeds in the direction of the arrow: the small disk rotates $90 ^ { \circ }$ per second. When the small disk completes one full rotation, the large disk rotates $90 ^ { \circ }$.
For example; 10 seconds after the beginning, the following view is obtained in the device and the indicator appears as $\square$.
What is the appearance of the indicator 100 seconds after the beginning?
(See answer choices A)–E) with figures in original paper.)

$A B C D E F$ regular hexagon
ABGH square
$[ \mathrm { AG } ] \cap [ \mathrm { BE } ] = \{ \mathrm { K } \}$
$\mathrm { m } ( \widehat { \mathrm { AKE } } ) = \mathrm { x }$
According to the given information above, what is $x$ in degrees?
A) 85 B) 90 C) 95 D) 100 E) 105

ABCD is a square
$\mathrm { AE } \perp \mathrm { EB }$
$| \mathrm { AB } | = 25 \mathrm {~cm}$
$| \mathrm { BE } | = 20 \mathrm {~cm}$
$| \mathrm { DE } | = \mathrm { x }$
According to the given information above, what is x in cm?
A) $8 \sqrt { 6 }$ B) $12 \sqrt { 2 }$ C) $6 \sqrt { 5 }$ D) $5 \sqrt { 10 }$ E) $10 \sqrt { 3 }$

The paper in the shape of an isosceles right triangle ABC shown in the figure is folded along [AD] so that side $AB$ falls onto side $AC$.
Accordingly, what is the ratio $\frac { | C D | } { | A B | }$?
A) $\frac { 1 } { 2 }$ B) $\frac { 2 } { 3 }$ C) $\frac { \sqrt { 2 } } { 2 }$ D) $2 - \sqrt { 2 }$ E) $3 - 2 \sqrt { 2 }$

A right circular cylinder with height 7 cm and completely filled with water and an empty right cone with the same base and height h cm are combined as shown in Figure 1.
When this solid is inverted as shown in Figure 2, the height of water inside the solid is 11 cm. What is h in cm?
A) 5 B) 5.5 C) 6 D) 6.5 E) 7

A rectangular prism made of unit cubes with edge lengths 2 units, 3 units, and 4 units has all its faces painted. Then, this prism is separated into 24 unit cubes.
In the final state, what is the total number of unpainted faces of these unit cubes?
A) 78 B) 82 C) 86 D) 90 E) 92

$$\left( \frac { 8 } { 3 } - \frac { 9 } { 4 } \right) \left( 4 + \frac { 4 } { 5 } \right)$$
What is the result of this operation?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 4 } { 3 }$
D) 1
E) 2

$$\frac { 8 ! - 7 ! - 6 ! } { 8 ! }$$
What is the result of this operation?
A) $\frac { 4 } { 5 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 6 } { 7 }$
D) $\frac { 7 } { 8 }$
E) $\frac { 8 } { 9 }$

$$\begin{array} { r } A C B \\ + \quad A C \\ \hline 3 B C \end{array}$$
According to this operation, what is the product $A \cdot C$?
A) 12
B) 14
C) 15
D) 16
E) 21

$$\begin{aligned} & \frac { a + c } { b } = \frac { 3 } { 2 } \\ & \frac { b } { c } = \frac { 3 } { 4 } \end{aligned}$$
Given this, what is the ratio $\frac { a } { b }$?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 5 } { 6 }$

Let $p$ be a prime number and $n$ be a natural number such that
$$p \cdot n = 3 ^ { p }$$
Given this equality, what is the sum $p + n$?
A) 10
B) 12
C) 15
D) 16
E) 18

An operation is defined on the set of positive real numbers as
$$a \bullet b = \frac { a \cdot b } { a + b }$$
Given that
$$\frac { 1 } { 2 } \bullet \frac { 3 } { 4 } = 3 \bullet \frac { 1 } { x }$$
what is $x$?
A) $\frac { 3 } { 2 }$
B) $\frac { 9 } { 4 }$
C) 2
D) 3
E) 5

Let $x$, $y$, and $z$ be integers such that

• The product $x \cdot y$ is an even number
• The sum $x + z$ is an odd number
• The sum $y + z$ is an odd number

Given this; I. $x$ is an odd number. II. $y$ is an even number. III. $z$ is an odd number. Which of these statements are true?
A) Only I
B) Only III
C) I and II
D) II and III
E) I, II and III