$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
What is the sum of the real numbers $x$ that satisfy the equation?
A) $\frac{2}{3}$
B) $\frac{3}{4}$
C) $\frac{3}{5}$
D) $\frac{5}{6}$
E) $\frac{7}{6}$

Given that $x - 2 y = 3$, what is the value of
$$x ^ { 2 } + 4 y ^ { 2 } - 4 x y - 2 y + x - 3$$
?
A) 4
B) 5
C) 8
D) 9
E) 15

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 3 } - 3 x ^ { 2 } y = 3 \\ & y ^ { 3 } - 3 x y ^ { 2 } = 11 \end{aligned}$$
Accordingly, what is the difference $x - y$?
A) 3
B) 2
C) 1
D) - 2
E) - 3

$$\frac { 2 ( x - y ) } { x - y - 1 } + \frac { x - y - 1 } { x - y - 2 } = 3$$
Given this, what is the difference $x - y$?
A) $\frac { - 1 } { 2 }$
B) $\frac { - 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 5 } { 3 }$
E) $\frac { 5 } { 4 }$

$$x \cdot \left( \sqrt { \frac { 1 } { x } - \frac { 1 } { x ^ { 2 } } } \right) = \frac { 1 } { 2 }$$
Given that, what is x?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 4 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 6 } { 5 }$
E) $\frac { 7 } { 5 }$

Let a and b be positive integers. The sum of the coefficients of the polynomial
$$P ( x ) = ( x + a ) \cdot ( x + b )$$
is 15. What is the sum $a + b$?
A) 10
B) 9
C) 8
D) 7
E) 6

$$x ^ { 2 } - ( \sin a ) x - \frac { 1 } { 4 } \left( \cos ^ { 2 } a \right) = 0$$
One root of the equation is $\frac { 2 } { 3 }$. Accordingly, what is $\sin a$?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 2 } } { 3 }$
C) $\frac { \sqrt { 2 } } { 6 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$

Let $\mathbf { a }$ and $\mathbf { b }$ be real numbers such that
$$\begin{aligned} & a ^ { 2 } - a = b ^ { 2 } - b \\ & a \cdot b = - 1 \end{aligned}$$
Given this, what is the sum $a ^ { 2 } + b ^ { 2 }$?
A) 6
B) 5
C) 4
D) 3
E) 2

$$a \cdot b = \frac { 3 } { 2 }$$
Given that, what is the value of the expression $\left( a + \frac { 1 } { 2 b } \right) \left( b - \frac { 1 } { a } \right)$?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 2 } { 5 }$

$a$ is a real number and
$$\left( 1 - a + a ^ { 2 } \right) \left( \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a ^ { 3 } } \right) = 9$$
Given this, what is $a$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 1 } { 4 }$

For distinct positive real numbers $x$ and $y$,
$$\left( \frac { x } { y } - \frac { y } { x } \right) \cdot \frac { x y } { 4 } = ( x - y ) ^ { 2 }$$
Given that, what is the ratio $\frac { x } { y }$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

Given that $a ^ { 2 } + a = 1$,
$$a ^ { 4 } - 2$$
Which of the following is the equivalent of this expression in terms of $a$?
A) $- a$
B) $- a + 2$
C) $- 2 a$
D) $- 2 a + 1$
E) $- 3 a$

$\frac { \mathrm { a } } { 5 } , \frac { \mathrm {~b} } { \mathrm { a } }$ and $\frac { \mathrm { a } } { 3 }$ are three consecutive integers arranged from smallest to largest.
Given this, what is the sum $\mathrm { a } + \mathrm { b }$?
A) 60
B) 70
C) 75
D) 80
E) 90

Let k be a positive real number. If one root of the equation
$$3 x ^ { 2 } + k x - 2 = 0$$
is k, what is the other root?
A) $\frac { \sqrt { 2 } } { 3 }$
B) $\frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { - 2 \sqrt { 2 } } { 3 }$
D) $\frac { - \sqrt { 2 } } { 6 }$
E) $\frac { - \sqrt { 3 } } { 6 }$

$$\frac { \sqrt { 2 - 2 x } } { \sqrt { 3 + 3 x } } = \frac { 1 } { 2 }$$
Given that this holds, what is x?
A) $\frac { 2 } { 7 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 4 } { 9 }$
D) $\frac { 5 } { 11 }$
E) $\frac { 7 } { 12 }$

$$\frac { \frac { 4 } { 3 } + \frac { 3 } { 4 } } { \frac { 2 } { 3 } - \frac { 1 } { 4 } }$$
What is the result of this operation?
A) 5
B) 10
C) 15
D) 20
E) 25

Let $\mathrm { a } , \mathrm { b }$ and c be prime numbers such that
$$\mathrm { ab } + \mathrm { ac } = 4 \mathrm { a } ^ { 2 } + 8$$
Given this, what is the product $\mathbf { a } \cdot \mathbf { b } \cdot \mathbf { c }$?

$$\lim _ { x \rightarrow 0 ^ { + } } ( \sin x ) \cdot ( \ln x )$$
Which of the following is this limit equal to?
A) $- 1$
B) 0
C) 1
D) $\infty$
E) $- \infty$

The sum of the prime divisors of a natural number A is;

• 3 less than the sum of the prime divisors of $12 \cdot A$.
• 5 less than the sum of the prime divisors of $70 \cdot A$.

Accordingly, what is the sum of the digits of the smallest value that A can take?
A) 4
B) 5
C) 6
D) 7
E) 8

The smaller of two numbers is 3 less than the arithmetic mean of these two numbers, and the larger is 4 more than the geometric mean of these two numbers.
Accordingly, what is the sum of these two numbers?
A) 7
B) 9
C) 10
D) 12
E) 14

The numbers $\frac { x } { y }$, $x - y$, and $x$ are three consecutive even integers arranged from smallest to largest.\ Accordingly, what is the sum $\mathrm{x} + \mathrm{y}$?\ A) 8\ B) 10\ C) 12\ D) 14\ E) 16

For positive real numbers $\mathrm{a}$, $\mathrm{b}$, and $c$ $$\begin{aligned}& \frac { a + c } { b + 2 } = \frac { c } { b } \\& \frac { a } { b } = c\end{aligned}$$ the following equalities are given.\ Accordingly, what is b?\ A) $\sqrt { 2 }$\ B) $\sqrt { 3 }$\ C) $\sqrt { 6 }$\ D) 2\ E) 3

A function $f$ on the set of real numbers is defined for every real number $x$ where $n$ is an integer as $$f ( x ) = x - n , \quad x \in [ n , n + 1 )$$ Accordingly, $$f ( 1 ) + f \left( \frac { 7 } { 3 } \right) + f \left( \frac { 13 } { 6 } \right)$$ what is this sum?\ A) $\frac { 1 } { 2 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 7 } { 6 }$\ D) 1\ E) 2

Let $n$ and $k$ be positive integers. The value of $n _ { k }$ is defined as
- If $n$ is divisible by $k$, then $n _ { k } = \frac { n } { k }$ - If $n$ is not divisible by $k$, then $n _ { k } = 0$
Example: $$\begin{aligned} & 10 _ { 2 } = 5 \\ & 10 _ { 3 } = 0 \end{aligned}$$
Accordingly,
$$n _ { 2 } + n _ { 3 } = 10$$
what is the sum of the $n$ numbers that satisfy the equality?
A) 24 B) 28 C) 32 D) 36 E) 40

A second-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients whose leading coefficient is 1 has two distinct roots that are $P ( 0 )$ and $P ( - 1 )$. Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 5 } { 2 }$
D) 1
E) 2