OAEF is a rectangle, ABCD is a square $| \mathrm { FE } | = 7$ units $| \mathrm { AB } | = 2$ units $| \mathrm { DE } | = x$
In the figure, points E and C are on a quarter circle with center O.
Accordingly, what is $x$ in units?
A) $\frac { 7 } { 2 }$ B) $\frac { 9 } { 2 }$ C) $\frac { 13 } { 4 }$ D) 3 E) 4

$$6 | \mathrm { AB } | = 3 | \mathrm { BC } | = 2 | \mathrm { CD } |$$
Above, three semicircles with diameters $[ \mathrm { AB } ] , [ \mathrm { BC } ]$ and $[ \mathrm { CD } ]$ with collinear centers are drawn inside a semicircle with diameter [AD], and the region between them is painted as shown in the figure.
If the perimeter of the painted region is $\mathbf { 24 \pi }$ units, what is its area in square units?
A) $44 \pi$ B) $48 \pi$ C) $52 \pi$ D) $56 \pi$ E) $60 \pi$

A square frame made by assembling four wires of equal length and fixed to the wall with nails at its corners as shown in Figure 1 covers an area of 100 square units on the wall.
As a result of the nails on corners A and B coming loose, one side slides down to form a rhombus shape as shown in Figure 2. In this frame, the height of corners A and B from the ground has decreased by 6 units each, while the position of the other two corners has not changed.
Accordingly, by how many square units has the area covered by the frame on the wall decreased?
A) 18 B) 20 C) 26 D) 30 E) 32

In the Cartesian coordinate plane, two circles with one centered at $(12,0)$ and the other centered at $(0,9)$ intersect only at point $(4,6)$.
What is the distance between the points on these circles that are closest to the origin?
A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $\sqrt { 13 }$ D) $2 \sqrt { 5 }$ E) $2 \sqrt { 10 }$

Identical boards in the shape of an isosceles trapezoid are joined together as shown in the figure to form a rectangular frame with a short side of 16 cm and a long side of 26 cm on the outside.
A picture is placed inside the frame of this frame such that the entire picture is visible and completely covers the inside of the frame. Accordingly, what is the area of this picture placed in $\mathbf { c m } ^ { \mathbf { 2 } }$?
A) 336
B) 312
C) 288
D) 264
E) 240

A point selected inside a pentagon is connected to the midpoints of the sides of the pentagon and to one vertex as shown in the figure. In this case, the regions formed are painted in different colors and the areas of these regions are written in square units on the figure.
According to this, what is the difference A - B?
A) 1
B) 1.5
C) 2
D) 2.5
E) 3

In the rectangular coordinate plane, a circle divided into two equal parts by the line $x + y = 4$ intersects the x-axis at a single point and the y-axis at two different points. Given that the distance between the points where the circle intersects the y-axis is 4 units, what is the circumference of the circle in units?
A) $4 \pi$
B) $5 \pi$
C) $6 \pi$
D) $7 \pi$
E) $8 \pi$

In a plane, three circles with radius r are constructed with the vertices of a right triangle $ABC$ as centers, and these circles do not intersect each other. The lengths of the parts on the sides of the triangle that are not inside these circles are given as 2 units, 3 units, and 5 units. Accordingly, what is the total area of the regions inside the circles but outside the triangle in square units?
A) $6 \pi$
B) $8 \pi$
C) $9 \pi$
D) $\frac { 9 \pi } { 2 }$
E) $\frac { 15 \pi } { 2 }$

In the figure, a semicircle with center A and radius $[AC]$ and a semicircle with center B and radius $[BC]$ are given. Point B is on the circle centered at A, and point A is on the circle centered at B.
Accordingly, what is the area of the shaded region in square units?
A) $36\pi$
B) $42\pi$
C) $48\pi$
D) $54\pi$
E) $60\pi$

In the rectangular coordinate plane, the line $y = m x$
$$x ^ { 2 } - 26 x + y ^ { 2 } + 144 = 0$$
intersects the circle at two different points.
Accordingly, which of the following is the interval showing all possible values of $m$?
A) $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$
B) $\left( - \frac { 3 } { 8 } , \frac { 3 } { 8 } \right)$
C) $\left( - \frac { 4 } { 9 } , \frac { 4 } { 9 } \right)$
D) $\left( - \frac { 5 } { 12 } , \frac { 5 } { 12 } \right)$
E) $\left( - \frac { 7 } { 24 } , \frac { 7 } { 24 } \right)$

Zeynep, who wants to prepare a cargo package, takes a right prism shaped cardboard box with a square base and a lid on its top surface as shown in Figure 1.
After placing what she wants to send in the box, Zeynep uses two blue colored bands, each with a width of 1 unit, to close the box. These two bands are parallel to the edges of the prism as shown in Figure 2, and each completely wraps around two side faces and the top face, excluding the bottom face. The total area covered by the bands on the surfaces of the prism is 25 square units.
Given that the total area of the outer surface of this box is 182 square units, what is the volume of the box in cubic units?
A) 100 B) 108 C) 147 D) 192 E) 196

In a rectangular coordinate plane, point $A(11, 9)$ is located in the interior of a circle that is tangent to the line $y = x$ at point $B(7, 7)$.
Accordingly, what is the smallest integer value that the radius of this circle can take in units?
A) 6 B) 8 C) 10 D) 12 E) 14

An equilateral triangle with red-colored sides and an equilateral triangle with blue-colored sides are drawn such that one vertex of each is on a side of the other triangle, as shown in the figure.
In the resulting figure, the area of the yellow-colored triangle equals 4 times the area of the gray-colored triangle.
Accordingly, what is the ratio of the area of the triangle with red-colored sides to the area of the triangle with blue-colored sides?
A) $\frac{2}{3}$ B) $\frac{5}{6}$ C) $\frac{8}{9}$ D) $\frac{25}{27}$ E) $\frac{25}{36}$

Two identical blue ropes have one end each tied to two nails on a wall at equal heights from the ground and 48 units apart. Then a circular plate is hung on these ropes such that the other ends of the ropes are attached to two points on the circumference of the plate and the ropes are perpendicular to the ground, as shown in Figure 1. Later, one of these ropes broke and the plate hung on the remaining rope, and when the rope is perpendicular to the ground, the view in Figure 2 is obtained, and the height of the plate from the ground decreased by 16 units compared to the initial situation.
Accordingly, what is the radius of this plate in units?
A) 25 B) 26 C) 29 D) 30 E) 32

In the figure, point $C$ is on the line segment $[AB]$, point $D$ is on the semicircle with diameter $[AB]$, and $m(\widehat{BAD}) = 18^{\circ}$.
In the figure, the area of the yellow-colored region equals 4 times the area of the blue-colored region. Accordingly, what is the ratio $\frac{|AC|}{|BC|}$?
A) $\frac{3}{2}$ B) $\frac{5}{3}$ C) $\frac{7}{4}$ D) $\frac{7}{5}$ E) $\frac{9}{5}$

A circle drawn in the rectangular coordinate plane

• has one common point with the line $d_{1}: y - \frac{4x}{3} - 46 = 0$,
• has two common points with the line $d_{2}: y - \frac{4x}{3} - 6 = 0$,
• has no common points with the line $d_{3}: y - \frac{4x}{3} - 1 \geqslant 0$.

It is known that. Accordingly, which of the following could be the radius of this circle in units?
A) 11 B) 13 C) 15 D) 17 E) 19

Let $m$ and $n$ be real numbers. In the rectangular coordinate plane, a circle passing through point $A(4, 1)$ is drawn with equation
$$x^{2} + y^{2} - 2x + 6y = n$$
The line $y = mx$ drawn in the plane intersects this circle at points $B$ and $C$. Given that $m(\widehat{BAC}) = 90^{\circ}$, what is the sum $m + n$?
A) 8 B) 9 C) 10 D) 11 E) 12

For research laboratories planned to be established on the Luna planet, two completely closed buildings with the same radii and volumes are designed to be placed on the ground as shown in the figure: one in the shape of a half right circular cylinder and the other in the shape of a hemisphere.
Accordingly, what is the ratio of the surface area of the half right circular cylinder building (excluding the ground) to the surface area of the hemisphere building (excluding the ground)?
A) $\frac{1}{2}$ B) $\frac{3}{5}$ C) $1$ D) $\frac{7}{6}$ E) $\frac{4}{3}$