ABCD is a trapezoid $\mathrm { DC } / / \mathrm { AB }$ DE//CB DE is an angle bisector $\mathrm { m } ( \widehat { \mathrm { DAE } } ) = 70 ^ { \circ }$ $\mathrm { m } ( \widehat { \mathrm { BCD } } ) = \mathrm { x }$ According to the given information above, what is x in degrees?
A) 105
B) 110
C) 115
D) 120
E) 125

A piece of paper in the shape of rectangle $ABCD$ has a point E marked on side $DC$ and a point F marked on side AB. When this paper is folded along line EF, AF and EC intersect perpendicularly as shown in the figure.
Given that the area of the figure obtained after the folding operation is $\mathbf { 18 }$ square units less than the area before the folding operation, what is the length |AD| in units?
A) 3
B) 4
C) 6
D) 8
E) 9

A square with side length a units is divided into a total of eight squares, seven of which are congruent, and the resulting large square is called $\mathrm { K } _ { 1 }$ (Figure 1). Then the square $\mathrm { K } _ { 1 }$ is similarly divided to obtain square $\mathrm { K } _ { 2 }$ (Figure 2). When square $\mathrm { K } _ { 2 }$ is similarly divided, the resulting square $\mathrm { K } _ { 3 }$ has a side length of 27 units.
Accordingly, what is a?
A) 36
B) 49
C) 64
D) 81
E) 100

The method of drawing a tangent to a circle with center O from an external point P is given below.

• Line segment OP is drawn.
• The midpoint M of line segment OP is determined.
• A circle with center M and diameter [OP] is drawn.
• The intersection points of the circles with centers O and M are marked. Let one of these points be T.
• Ray [PT is tangent to the circle with center O at point T.

In this construction, if the radii of the circles with centers O and M are 4 cm and 3 cm respectively, what is the length $| PT |$ in cm?
A) $3 \sqrt { 3 }$
B) $2 \sqrt { 5 }$
C) $\sqrt { 7 }$

Let m, n be non-zero real numbers. The function
$$f ( x ) = \frac { x } { n } \sin \left( \frac { m } { x } \right)$$
has a horizontal asymptote at $y = 2$. Given this, which of the following is the relationship between m and n?
A) $m = n$
B) $m = n + 2$
C) $m = 2 n$
D) $m = 3 n$
E) $2 m = 3 n$

Two tanks in the shape of right circular cylinders with equal heights and base radii of 2 meters and 3 meters respectively are initially empty. Two separate faucets that discharge the same amount of water per unit time are used; the faucet for the small tank is opened 5 minutes after the faucet for the large tank is opened. When the faucet for the small tank is opened, the height of water in the large tank is 2 meters.
Accordingly, how many minutes after water starts being supplied to the small tank will the water heights in the tanks be equal?
A) 2
B) 3
C) 4
D) 5
E) 6

Below is the graph of the function f. If the function $( \mathbf { f } + \mathbf { g } )$ is continuous at the point $X = 1$, which of the following could be the graph of the function g?
A) [graph A]
B) [graph B]
C) [graph C]
D) [graph D]
E) [graph E]

$ABCDEF$ is a regular hexagon $|AL| = |LB|$ $| \mathrm { BC } | = 4 \mathrm {~cm}$ $| \mathrm { DK } | = 3 \mathrm {~cm}$ $|KE| = \mathrm { x }$ According to the given information above, what is x in cm?
A) $4 \sqrt { 3 }$
B) $3 \sqrt { 5 }$
C) $3 \sqrt { 7 }$
D) 6
E) 7

$$\left( 1 - \frac { 3 } { 5 } \right) \left( 1 - \frac { 3 } { 8 } \right) \left( 1 - \frac { 5 } { 13 } \right)$$
What is the result of this operation?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 5 }$
C) $\frac { 5 } { 8 }$
D) $\frac { 2 } { 13 }$
E) $\frac { 8 } { 13 }$

$$0,75 - \frac { 0,2 } { 0,3 + \frac { 0,1 } { 0,5 } }$$
What is the result of this operation?
A) 0.25
B) 0.35
C) 0.45
D) 0.5
E) 0.6

Let $p , q , r$ be prime numbers with
$$2 < p < q < r < 15$$
Accordingly, how many different values can the product $p \cdot q \cdot r$ take?
A) 4
B) 6
C) 8
D) 10
E) 12

For every positive integer $n$, the number $n$ is defined as
$$n = ( n ) ( n + 2 ) ( n + 4 ) ( n + 6 )$$
Given this, what is the value of the expression $\frac { 12 - 10 } { 8 }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 3 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 8 } { 5 }$

$a , b , c , d$ are distinct real numbers and
$$\begin{aligned} & b + c = d \\ & a \cdot b \cdot c = 0 \end{aligned}$$
Which of the following is true?
A) $a = 0$
B) $b = 0$
C) $c = 0$
D) $a+c=0$
E) $a + d = 0$

When the two-digit natural numbers AB and BA are divided by 17, the sum of the remainders obtained is 17.
Given this, what is the value of $| A - B |$?
A) 1
B) 2
C) 3
D) 4
E) 5

$a$ is a positive integer and
$$\operatorname { LCM } ( 5 , a ) = \operatorname { GCD } ( 20 , a )$$
Given this, what is the sum of the values that $a$ can take?
A) 25
B) 30
C) 35
D) 40
E) 45

Let $n$ be a positive integer such that the remainder when 33 is divided by $n$ is 5.
Given this, what is the sum of the values that $n$ can take?
A) 42
B) 44
C) 45
D) 48
E) 49

Let $X , Y$ and $Z$ be sets. The following proposition is given:
$$\text { " If } ( X \subseteq Y \text { and } X \subseteq Z ) \text { then } Y \subseteq Z \text {." }$$
Which of the following is a counterexample that shows this proposition is false?

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

A function $f$ defined on the set of integers satisfies the following equalities for every integer $n$:
$$\begin{aligned} & f ( n + 2 ) = f ( n ) + 4 \\ & f ( n + 3 ) = f ( n ) + 6 \end{aligned}$$
Given that $f ( 4 ) = 5$, what is the value of $f ( 11 )$?
A) 15
B) 17
C) 19
D) 21
E) 23

Sets $A = \{ 1,2,3 \}$ and $B = \{ 2,3,4,5 \}$ are given.
Given this, how many functions $\mathbf { f } : \mathbf { A } \rightarrow \mathbf { B }$ can be defined such that for every $a \in A$
$$a + f ( a ) \leq 6$$
A) 12
B) 18
C) 20
D) 24
E) 27

All three-element subsets of a four-element set A whose elements are integers are written out. When the arithmetic mean of the elements of each of these subsets is calculated, the values 8, 9, 10, and 11 are found.
Given this, which of the following is not an element of set A?
A) 5
B) 8
C) 11
D) 14
E) 17

If the difference between two prime numbers $p$ and $q$ is 4, then the pair $(p , q)$ is called a "cousin prime pair."
Given this, which of the following cannot be the sum of a cousin prime pair?
A) 18
B) 30
C) 42
D) 66
E) 78

Let $x , y$ be integers such that
$$\begin{aligned} & 0 < x < 100 \\ & 0 < y < 100 \end{aligned}$$
Given this, for how many ordered pairs $(x , y)$ is the sum $x + y$ a three-digit number?
A) 1980
B) 2500
C) 4500
D) 4950
E) 5050

In a bakery, 40 sesame rings and 50 pastries are sold for a total of 100 TL. A sesame ring vendor gives 100 TL to the baker for 30 sesame rings and 50 pastries and receives A TL in change.
What is the total price of 1 sesame ring and 1 pastry in terms of A in TL?
A) $\frac { A + 20 } { 10 }$
B) $\frac { \mathrm { A } + 50 } { 10 }$
C) $\frac { A + 100 } { 50 }$
D) $\frac { 100 - A } { 50 }$
E) $\frac { 100 - A } { 50 - A }$

Fresh grapes with a water content of 36% by weight are left to dry. After some time, the water content in these grapes becomes 8% by weight.
What is the weight of the grapes after this time in kg?
A) 12
B) 13
C) 14
D) 15
E) 16