Let $a$ and $b$ be positive real numbers. In the Cartesian coordinate plane, using the parabola
$$p ( x ) = ( x - a ) ^ { 2 } - b$$
that passes through the origin, three parabolas defined as
$$\begin{aligned} & p ( x + a ) + b \\ & p ( x + a ) - b \\ & p ( x - a ) - b \end{aligned}$$
have their vertices at the vertices of a triangle with an area of 16 square units.
Accordingly, what is the sum $a + b$?
A) 6 B) 9 C) 12 D) 15 E) 18

Let $0 < x _ { 1 } < x _ { 2 }$. A function f defined on the set of real numbers as
$$f ( x ) = \left( x - x _ { 1 } \right) \left( x - x _ { 2 } \right)$$
The parabola represented by this function intersects the axes at different points A and B in the rectangular coordinate plane as shown in the figure.
The distances from points A and B to the origin are equal, and this parabola takes its minimum value when $x = \frac { 3 } { 5 }$. Accordingly, what is the ratio $\frac { \mathbf { x } _ { \mathbf { 2 } } } { \mathbf { x } _ { \mathbf { 1 } } }$?
A) 2
B) 3
C) 4
D) 5
E) 6

Where $a, b$ and $c$ are real numbers,
$$y = ax^2 + bx + c$$
the parabola intersects the line $y = 1$ at points B and C, and intersects the line $y = 6$ at only point A. The locations of points A, B and C in the rectangular coordinate plane are shown in the figure below.
Accordingly, what are the signs of the numbers $a$, $b$ and $c$ respectively?
A) +, -, -
B) +, +, -
C) -, +, +
D) -, -, +
E) -, -, -

Ayşe, who wants to set aside a portion of her notebook for history class, folds the page she wants to set aside more easily to find, as shown in the figure, from the upper right corner of the page.
In this notebook with rectangular pages, each line on the pages is parallel to the top edge of the page.
Accordingly, what is x in degrees?
A) 50 B) 52 C) 54 D) 56 E) 58

Three right triangles are positioned as follows: their hypotenuses lie on the same line and one vertex of each coincides.
Given that each of the blue angles measures $115^{\circ}$, what is the measure of the yellow angle in degrees?
A) 100 B) 105 C) 110 D) 115 E) 120

A glazier using his extendable ladder to reach a window 6 meters high from the ground positions the ladder 4.8 meters away from a garden wall 3.6 meters high, as shown in Figure 1, and extends the ladder to touch the wall and reach the bottom of the window. The glazier then brings the ladder into the garden and, as shown in Figure 2, leans one end against the wall and extends it to the bottom of the window.
When the glazier positions the ladder as shown in Figure 2, he extends it 3.5 meters less than when he positions it as shown in Figure 1. Accordingly, what is the thickness of the wall in meters?
A) 0.5 B) 0.6 C) 0.7 D) 0.8 E) 0.9

In an application used to adjust the sound level of a music program, consisting of 100 equal units in the shape of a right triangle, the appearance of the application when the sound level is 60 units is given in Figure 1.
When the sound level is increased to 70 units as shown in Figure 2, the area of the green right triangle increases by 260 square units.
Accordingly, what is the height marked with ? in the appearance of the application in units?
A) 30 B) 35 C) 40 D) 45 E) 50

Three square-shaped tables with perimeters of 12, 16, and 28 units are given in Figure 1. These three tables are combined as shown in Figure 2 with no gaps between them to create a new table.
Accordingly, what is the perimeter length of the new table created in units?
A) 42 B) 46 C) 48 D) 52 E) 54

An isosceles trapezoid-shaped cardboard with a height of 30 units has an upper base length of 20 units. When this cardboard is cut along a line parallel to the lower base, reducing the height by 12 units, it is observed that the lower base length decreases by 6 units.
Accordingly, what is the area of the new cardboard obtained in square units?
A) 363 B) 385 C) 441 D) 450 E) 464

The interior angle measure of a regular n-sided polygon is calculated as $\frac{(n-2) \cdot 180^{\circ}}{n}$.
Six identical isosceles trapezoid-shaped mirrors, each with a perimeter of 28 units and shown in Figure 1, are combined as shown in Figure 2 with no gaps between them and all mirrors visible. In the resulting figure, the sum of the perimeter lengths of the red regular hexagon and the blue regular hexagon is 96 units.
Accordingly, what is the area of one of the mirrors used in square units?
A) $18\sqrt{3}$ B) $24\sqrt{3}$ C) $28\sqrt{3}$ D) $30\sqrt{3}$ E) $36\sqrt{3}$

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = \begin{cases} x^{2} - ax + 6 & , x \leq a \\ 2x + a & , a < x \leq b \\ 11 - 2x + b & , x > b \end{cases}$$
is continuous on its domain.
Accordingly, what is the product $a \cdot b$?
A) 4 B) 6 C) 8 D) 10 E) 12