$$\frac { 1 + \sqrt { a } } { 1 - a } - \frac { a } { 1 - \sqrt { a } } = \frac { 5 } { 3 }$$
Given this, what is a?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 9 }$
E) $\frac { 4 } { 9 }$

$$\left[ \left( \frac { - 1 } { 2 } \right) ^ { - 2 } \right] ^ { 3 }$$
What is the result of this operation?
A) - 32
B) - 16
C) 12
D) 32
E) 64

$6 ^ { - 3 } \left( 8 ^ { 3 } - 6 ^ { 3 } + 4 ^ { 3 } \right)$ What is the result of this operation?
A) $\frac { 1 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 5 } { 6 }$

For a real number $\mathbf { x }$,
$$\sqrt { \frac { \sqrt { x } + 2 } { \sqrt { x } - 2 } } = \sqrt { x } + 2$$
Given that, what is x?
A) 3
B) 4
C) 5
D) 6
E) 7

Given that $x \neq - 1$ and
$$( x + 1 ) \sqrt { x - 1 } = \sqrt { 2 x + 2 }$$
What is $x$?
A) $\frac { 3 \sqrt { 3 } } { 2 }$
B) $\frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { 4 \sqrt { 2 } } { 3 }$
D) $\sqrt { 2 }$
E) $\sqrt { 3 }$

$$3 ^ { 2 } \cdot \frac { 1 - 3 ^ { - 4 } } { 1 - 3 ^ { - 2 } }$$
What is the result of this operation?
A) 4
B) 5
C) 8
D) 9
E) 10

$$\frac { \sqrt [ 3 ] { 2 \cdot \sqrt { 54 } } } { \sqrt { 2 } }$$
What is the result of this operation?
A) $\sqrt { 2 }$ B) $\sqrt { 3 }$ C) $\sqrt { 6 }$ D) $\sqrt [ 3 ] { 4 }$ E) $\sqrt [ 3 ] { 9 }$

$$\frac { 6 ^ { 4 } - 4 ^ { 4 } } { 5 \cdot 2 ^ { 4 } }$$
What is the result of this operation?
A) 9
B) 12
C) 13
D) 14
E) 16

$$\frac { 4 } { 9 - \frac { 49 } { 9 } } - \frac { 1 } { 8 }$$
What is the result of this operation?
A) 1
B) 2
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 4 }$

$$\frac { 6 ^ { -8 } \cdot 9 ^ { 4 } } { 4 ^ { -6 } }$$
What is the result of this operation?
A) 8
B) 9
C) 12
D) 16
E) 18

$$\frac { \sqrt { 48 } } { \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { \sqrt { 27 } } }$$
What is the result of this operation?
A) 3
B) 5
C) 8
D) 9
E) 12

$$\frac { \sqrt { 12 } } { \sqrt { 27 } + \frac { 1 } { \sqrt { 3 } } }$$
What is the result of this operation?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 5 }$
C) $\frac { 1 } { 2 }$
D) $\sqrt { 3 }$
E) $\sqrt { 6 }$

$$\begin{aligned} & a = \sqrt { 2 } + \sqrt { 45 } \\ & b = \sqrt { 5 } + \sqrt { 18 } \\ & c = \sqrt { 8 } + \sqrt { 20 } \end{aligned}$$
Given this, which of the following orderings is correct?
A) a $<$ b $<$ c
B) b $<$ a $<$ c
C) c $<$ b $<$ a
D) b $<$ c $<$ a
E) c $<$ a $<$ b

Positive real numbers $a$ and $b$ satisfy the equality
$$a ^ { 2 } - 2 a b - 3 b ^ { 2 } = 0$$
Accordingly, what is the value of the expression $\frac { a + b } { a - b }$?
A) 2
B) 3
C) 4
D) 5
E) 6

$$\frac { 1 } { \sqrt { 2 x } } + \frac { 4 } { \sqrt { 8 x } } = 6$$
Given this, what is $x$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 1 } { 8 }$
E) $\frac { 1 } { 12 }$

$\frac { 5 - \frac { 25 } { 9 } } { \frac { 2 } { 3 } } - \frac { 1 } { 3 }$\ What is the result of this operation?\ A) 1\ B) 2\ C) 3\ D) 4\ E) 5

$\frac { 60 ^ { 2 } \cdot 3 } { 15 ^ { 3 } }$\ What is the result of this operation?\ A) 2.4\ B) 2.6\ C) 2.8\ D) 3\ E) 3.2

$\frac { \sqrt { 48 } + \sqrt { 75 } } { \sqrt { 108 } - \sqrt { 27 } }$\ What is the result of this operation?\ A) 1\ B) 2\ C) 3\ D) 4\ E) 5

$$\sqrt [ 3 ] { \frac { 32 } { \sqrt { 8 } - \sqrt { 2 } } }$$
What is the result of this operation?
A) $\sqrt { 2 }$ B) $2 \sqrt { 2 }$ C) $\sqrt [ 3 ] { 2 }$ D) 2 E) 4

$\frac { 1 } { \sqrt { \mathrm{a} } } - \frac { 2 } { \sqrt { 9 \mathrm{a} } } = 1$\ Given this, what is a?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 1 } { 4 }$\ D) $\frac { 1 } { 9 }$\ E) $\frac { 4 } { 9 }$

For a project, 16 schools were selected from each of Turkey's 81 provinces, and a message was sent to each school's principal. Then, each school's principal sent this message to 35 teachers in their school.
Accordingly, what is the total number of principals and teachers to whom this message was sent?
A) $4^{6}$
B) $5^{6}$
C) $6^{6}$
D) $7 \cdot 5^{5}$
E) $8^{6}$

For integers $x$ and $y$
$$9 ^ { x } - 3 ^ { 2 x - 2 } = 2 ^ { y } \cdot 3 ^ { 6 }$$
the equality is satisfied. Accordingly, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) 6
E) 7

When the numbers $\sqrt{5}, \sqrt{8}, \sqrt{12}, \sqrt{18}, \sqrt{20}$ and $\sqrt{27}$ are placed in the boxes below, with each box containing a different number, A, B, and C become whole numbers.
Accordingly, what is the sum $\mathrm{A} + \mathrm{B} + \mathrm{C}$?
A) 40
B) 44
C) 48
D) 52
E) 56

For distinct natural numbers $a$, $b$, and $c$,
$$\frac { 6 ^ { a } \cdot 15 ^ { b } } { 9 ^ { b } \cdot 10 ^ { c } }$$
is equal to an integer. Accordingly, which of the following orderings is correct?
A) $a < b < c$
B) $b < a < c$
C) $b < c < a$
D) $c < a < b$
E) $c < b < a$

Mert, who performs operations with radical numbers, instead of multiplying the number $\sqrt{10} + \sqrt{6}$ by its conjugate $\sqrt{10} - \sqrt{6}$, mistakenly divided it.
Accordingly, how much greater is the number Mert found than the number he should have found?
A) $\sqrt{12}$ B) $\sqrt{15}$ C) $\sqrt{18}$ D) $\sqrt{20}$ E) $\sqrt{30}$