Teacher Cemal conducted the following activity step by step with his students in a geometry lesson and asked them a question at the end of the activity.

• Let us draw a line segment AB of length 8 cm.
• Let us open our compass to 5 cm.
• By placing the sharp point of the compass first at point A and then at point B, let us draw two circles.
• Let us name the intersection points of these two circles as C and D.
• Let us form the quadrilateral ACBD with vertices at points A, B, C, and D.

• What is the area of the quadrilateral region ACBD in $\mathrm { cm } ^ { 2 }$?

According to this, what is the answer to the question asked by Teacher Cemal?
A) 20
B) 24
C) 25
D) 26
E) 32

Let A, B and C be sets. I. If $A \cup B = A \cup C$ then $B = C ^ { \prime }$. II. If $\mathrm { A } \cap \mathrm { B } = \varnothing$ then $\mathrm { A } \backslash \mathrm { B } = \mathrm { A } ^ { \prime }$. III. If $A \cup B = A$ then $B \backslash A = \varnothing$. Which of these propositions are always true?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

A student made an error while proving the following claim that he thought was true.
Claim: Let $f : X \rightarrow Y$ be a function, and let $A$ and $B$ be subsets of $X$. Then $f ( A \cap B ) = f ( A ) \cap f ( B )$.
The student's proof: If I show that the sets $f ( A \cap B )$ and $f ( A ) \cap f ( B )$ are subsets of each other, the proof is complete.
Now let $c \in f ( A \cap B )$. I. There exists a $d \in A \cap B$ such that $c = f ( d )$. II. Since $d \in A$ and $d \in B$, we have $f ( d ) \in f ( A )$ and $f ( d ) \in f ( B )$. Thus $c = f ( d ) \in f ( A ) \cap f ( B )$.
On the other hand, let $c \in f ( A ) \cap f ( B )$. III. We have $c \in f ( A )$ and $c \in f ( B )$. From this, there exists an $a \in A$ such that $c = f ( a )$ and a $\mathrm { b } \in \mathrm { B }$ such that $c = f ( b )$. IV. Since $c = f ( a )$ and $c = f ( b )$, we have $a = b$. V. Since $a \in A , b \in B$ and $a = b$, we have $a \in A \cap B$ and thus $c = f ( a ) \in f ( A \cap B )$.
In which of the numbered steps did this student make an error?
A) I
B) II
C) III
D) IV
E) V

Ali; starting with the equality $x = y$ for non-zero, equal real numbers x and y, follows the following steps in order.
I. Let us multiply both sides of the equality by x: $$x ^ { 2 } = x \cdot y$$
II. Let us subtract $\mathrm { y } ^ { 2 }$ from both sides: $$x ^ { 2 } - y ^ { 2 } = x \cdot y - y ^ { 2 }$$
III. Let us factor both sides: $$( x + y ) ( x - y ) = y ( x - y )$$
IV. Let us divide both sides by $\mathrm { x } - \mathrm { y }$: $$x + y = y$$
V. Let us substitute y for x: $$2 y = y$$
As a result of these steps, Ali arrives at the conclusion "Every number equals twice itself."
Accordingly, in which of the numbered steps did Ali make an error?
A) I B) II C) III D) IV E) V

For propositions $p$, $q$, and $r$ $$( p \Rightarrow q ) \Rightarrow r$$ it is known that the proposition is false.
Accordingly,\ I. $p \Rightarrow q$\ II. $q \Rightarrow r$\ III. $r \Rightarrow p$\ Which of the following propositions are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and III\ E) II and III

A student made an error while proving the following claim that he believed to be true.
Claim: The number $\pi$ equals the number $e$.\ The student's proof: Let $f ( x )$ and $g ( x )$ be functions for $x > 0$ defined as $\mathrm{f} ( \mathrm{x} ) = \ln ( \pi \mathrm{x} )$ and $\mathrm{g} ( \mathrm{x} ) = \ln ( \mathrm{ex} )$.\ I. For every $x > 0$, the derivatives of functions $f ( x )$ and $g ( x )$ are equal to each other.\ II. Therefore, for every $x > 0$, functions $f ( x )$ and $g ( x )$ are equal to each other.\ III. Since $\ln ( x )$ is one-to-one and $f ( x ) = g ( x )$, we conclude that for every $x > 0$, $\pi x = ex$.\ IV. If two functions are equal for every $x > 0$, then their values at $x = 1$ are the same.\ V. Since the values of the functions $\pi \mathrm{x}$ and $ex$ at $x = 1$ are the same, we conclude that $\pi = \mathrm{e}$.\ In which of the numbered steps did this student make an error?\ A) I\ B) II\ C) III\ D) IV\ E) V

Let $a$ and $b$ be integers. The notation $\mathrm { a } \mid \mathrm { b }$ means that $a$ divides $b$ exactly.
A student wants to prove that the proposition "If integers $a$, $b$ and $c$ satisfy the conditions $a \mid c$ and $b \mid c$, then $(a + b) \mid c$ also holds." is false by using the counterexample method.
Accordingly, which of the following could be the example given by the student?

Regarding a two-digit natural number $AB$, the following propositions are given: p: The number $AB$ is even. q: The number $A^{AB}$ is prime. r: $A + B = 11$
If the proposition $(p \Rightarrow q) \wedge (q' \wedge r)$ is true, what is the product $A \cdot B$?
A) 18
B) 20
C) 24
D) 28
E) 30

The surface area of a rectangular prism with edge lengths $a, b$ and $c$ is calculated with the formula
$$A = 2(a \cdot b + a \cdot c + b \cdot c)$$
Two identical rectangular prisms are placed in three different ways such that they share one face each. The surface areas of the resulting Figure 1, Figure 2, and Figure 3 are calculated as 18, 20, and 22 square units respectively.
Accordingly, what is the surface area of one of the identical prisms in square units?
A) 12 B) 13 C) 14 D) 15 E) 16

Three faces of a square right prism-shaped board are painted white, and the other three faces are painted red. The sum of the areas of the white-painted faces is 76 square units, and the sum of the areas of the red-painted faces is 12 square units.
Accordingly, what is the volume of this board in cubic units?
A) 18 B) 24 C) 27 D) 32 E) 36

A toy bear, a toy horse, and a cactus plant are placed on 3 wall shelves, each at a different height from the ground, first as shown in Figure 1, and then as shown in Figure 2. The heights that are equal in Figure 1 and Figure 2 are shown with dashed lines. It is known that the sum of the heights of the toy bear, toy horse, and cactus plant is 15 units.
Given that the height of the leftmost shelf from the ground is 18 units, what is the sum of the heights of the other two shelves from the ground?
A) 45 B) 48 C) 51 D) 54 E) 57