For positive real numbers $x$ and $y$
$$\frac { x } { 8 } = \frac { y } { 12 } = \frac { 9 } { y - x }$$
Given this, what is the sum $x + y$?
A) 10
B) 15
C) 20
D) 25
E) 30

$$\frac { x ^ { 3 } - x ^ { 2 } y - x y ^ { 2 } + y ^ { 3 } } { 2 x ^ { 2 } - 4 x y + 2 y ^ { 2 } } = \frac { 1 } { 2 }$$
Given that, what is the sum $\mathbf { x } + \mathbf { y }$?
A) 1
B) 2
C) 4
D) $\frac { 3 } { 2 }$
E) $\frac { 4 } { 3 }$

Let $\mathbf { k }$ be a nonzero real number such that
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } = ( 6 k ) ^ { 2 } \\ & ( x - 2 k ) ^ { 2 } + y ^ { 2 } = ( 2 k \sqrt { 5 } ) ^ { 2 } \end{aligned}$$
Accordingly, which of the following is the equivalent of $x ^ { 2 } - y ^ { 2 }$ in terms of $k$?
A) $13 \mathrm { k } ^ { 2 }$
B) $14 \mathrm { k } ^ { 2 }$
C) $15 k ^ { 2 }$

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

In the Cartesian coordinate plane; a triangle with one vertex at the origin and the other vertices on the lines $y = x$ and $y = - x$ has its medians intersecting at point $(2,4)$.
Accordingly, what is the area of this triangle in square units?
A) 18 B) 24 C) 27 D) $9 \sqrt { 2 }$ E) $18 \sqrt { 2 }$

\c{C}\i{}nar has a total of 78 pens, some of which are blue. He distributed these pens among three pen holders as follows.

• The number of pens in the pen holders is directly proportional to 3, 4, and 6.
• The number of blue pens in each pen holder is equal to each other.
• In one of the pen holders, the ratio of the number of blue pens to the total number of pens in that holder is $\frac{1}{2}$; in another pen holder, this ratio is $\frac{1}{3}$.

Accordingly, how many blue pens does \c{C}\i{}nar have in total?
A) 18
B) 24
C) 27
D) 30
E) 36

Let $x$ and $y$ be positive real numbers such that
$$\begin{aligned} & x ^ { 2 } + 3 y ^ { 2 } = 8 \\ & 2 x ^ { 2 } + y ^ { 2 } = 6 \end{aligned}$$
What is the product $x \cdot y$?
A) 2
B) 4
C) 6
D) 8
E) 10

Let $x$ and $y$ be real numbers,
$$\begin{aligned} & x^{2} + 8xy = 60 \\ & y^{2} - 3xy = -15 \end{aligned}$$
Given that this holds, what is the product $\mathrm{x} \cdot \mathrm{y}$?
A) 3 B) 4 C) 5 D) 6 E) 7

For positive real numbers $a, x$ and $y$
$$\begin{aligned} & -2x^{2} + y^{2} = 2a \\ & 3x^{2} - 2y^{2} = -6a \end{aligned}$$
what is the ratio $\dfrac{y}{x}$?
A) 1 B) $\sqrt{2}$ C) $\sqrt{3}$ D) 2 E) 3