In coordinate space, the $xy$-plane, $yz$-plane, and $zx$-plane divide the space into 8 regions. How many of these 8 regions does the sphere
$$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 24$$
pass through? [4 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4

As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$.
When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]

In coordinate space, a sphere $S$ with center coordinates all positive has center at $(x, y, z)$ where $x > 0, y > 0, z > 0$, is tangent to the $x$-axis and $y$-axis respectively, and intersects the $z$-axis at two distinct points. The area of the circle formed by the intersection of sphere $S$ and the $xy$-plane is $64 \pi$, and the distance between the two intersection points with the $z$-axis is 8. What is the radius of sphere $S$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

In coordinate space, find the sum of all real numbers $k$ such that the plane $x + 8 y - 4 z + k = 0$ is tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 y - 3 = 0$. [3 points]

15. A quadrangular pyramid $P - A B C D$ has all vertices on the surface of sphere $O$. $PA$ is perpendicular to the plane containing rectangle $A B C D$. $AB = 3$, $AD = \sqrt { 3 }$. The surface area of sphere $O$ is $13 \pi$. The length of segment $PA$ is \_\_\_\_.

12. Given that the four vertices of tetrahedron $P - A B C$ lie on the surface of sphere $O$, with $P A = P B = P C$, $\triangle A B C$ is an equilateral triangle with side length 2, $E , F$ are the midpoints of $P A , A B$ respectively, and $\angle C E F = 90 ^ { \circ }$, then the volume of sphere $O$ is
A. $8 \sqrt { 6 } \pi$
B. $4 \sqrt { 6 } \pi$
C. $2 \sqrt { 6 } \pi$
D. $\sqrt { 6 } \pi$
II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

12. Given that the four vertices of tetrahedron $P - A B C$ lie on the surface of sphere $O$ , with $P A = P B = P C$ , $\triangle A B C$ is an equilateral triangle with side length 2, $E , F$ are the midpoints of $P A , A B$ respectively, and $\angle C E F = 90 ^ { \circ }$ , then the volume of sphere $O$ is
A. $8 \sqrt { 6 } \pi$
B. $4 \sqrt { 6 } \pi$
C. $2 \sqrt { 6 } \pi$
D. $\sqrt { 6 } \pi$
Section II: Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

Let $A , B , C$ be three points on the surface of sphere $O$, and $\odot O _ { 1 }$ be the circumcircle of $\triangle A B C$. If the area of $\odot O _ { 1 }$ is $4 \pi$ and $A B = B C = A C = O O _ { 1 }$ , then the surface area of sphere $O$ is
A. $64 \pi$
B. $48 \pi$
C. $36 \pi$
D. $32 \pi$

Given is the sphere with center $M ( 1 | 4 | 0 )$ and radius 6. (1a) [3 marks] Determine all values $p \in \mathbb { R }$ for which the point $P ( 5 | 1 | p )$ lies on the sphere. (1b) [2 marks] The line $g$ is tangent to the sphere at the point $B ( - 3 | 8 | 2 )$. Find a possible equation of $g$.
For each value of $a$ with $a \in \mathbb { R }$, a line $g _ { a }$ is given by $g _ { a } : \vec { X } = \left( \begin{array} { c } 2 \\ a - 4 \\ 4 \end{array} \right) + \lambda \cdot \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) , \lambda \in \mathbb { R }$
(2a) [2 marks] Determine, depending on $a$, the coordinates of the point where $g _ { a }$ intersects the $x _ { 1 } x _ { 2 }$ plane.
(2b) [3 marks] For exactly one value of $a$, the line $g _ { a }$ has an intersection point with the $x _ { 3 }$-axis. Find the coordinates of this intersection point.
On a playground, a triangular sun sail is erected to shade a sandbox. For this purpose, metal poles are fixed in the ground at three corners of the sandbox, at whose ends the sun sail is fastened. In a Cartesian coordinate system, the $x _ { 1 } x _ { 2 }$-plane represents the horizontal ground. The sandbox is described by the rectangle with corner points $K _ { 1 } ( 0 | 4 | 0 ) , K _ { 2 } ( 0 | 0 | 0 ) , K _ { 3 } ( 3 | 0 | 0 )$ and $K _ { 4 } ( 3 | 4 | 0 )$. The sun sail is represented by the planar triangle with corner points $S _ { 1 } ( 0 | 6 | 2,5 ) , S _ { 2 } ( 0 | 0 | 3 )$ and $S _ { 3 } ( 6 | 0 | 2,5 )$ (see Figure 1). One unit of length in the coordinate system corresponds to one meter in reality. [Figure]
The three points $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ determine the plane $E$. Sub-task Part B a (4 marks) Find an equation of the plane $E$ in normal form. (for verification: $E : x _ { 1 } + x _ { 2 } + 12 x _ { 3 } - 36 = 0$ ) Sub-task Part B b (3 marks) The manufacturer of the sun sail recommends stabilizing the metal poles used with additional safety cables if the sun sail area is more than $20 \mathrm {~m} ^ { 2 }$. Assess whether such stabilization is necessary in the present situation based on this recommendation.
Sunrays fall on the sun sail, which in the model and in Figure 1 can be represented by parallel lines with direction vector $\overrightarrow { S _ { 1 } K _ { 1 } }$. The sun sail casts a triangular shadow on the ground. The shadows of the corners of the sun sail designated by $S _ { 2 }$ and $S _ { 3 }$ are designated by $S _ { 2 } ^ { \prime }$ and $S _ { 3 } ^ { \prime }$ respectively.
Sub-task Part B c (2 marks) Justify without further calculation that $S _ { 2 } ^ { \prime }$ lies on the $x _ { 2 }$-axis. Sub-task Part B d (3 marks) $S _ { 3 } ^ { \prime }$ has the coordinates $( 6 | - 2 | 0 )$. Draw the triangle representing the shadow of the sun sail in Figure 1. Decide from the drawing whether more than half of the sandbox is shaded.
Sub-task Part B e (3 marks) To ensure the drainage of rainwater, the sun sail must have an inclination angle of at least $8 ^ { \circ }$ with respect to the horizontal ground. Justify that the drainage of rainwater is not ensured in the present case.
Sub-task Part B f (5 marks) In heavy rain, the sun sail deforms and sags. A so-called water pocket forms from rainwater that cannot drain away. The top surface of the water pocket is horizontal and is approximately circular with a diameter of 50 cm. At its deepest point, the water pocket is 5 cm deep. For simplicity, the water pocket is considered as a spherical segment (see Figure 2).
[Figure]
Fig. 2
The volume $V$ of a spherical segment can be calculated using the formula $V = \frac { 1 } { 3 } \pi h ^ { 2 } \cdot ( 3 r - h )$, where $r$ denotes the radius of the sphere and $h$ denotes the height of the spherical segment. Determine how many liters of water are in the water pocket.

The line segment $[ \mathrm { PQ } ]$ with endpoints $P ( 8 | - 5 | 1 )$ and $Q$ is a diameter of a sphere with center $M ( 5 | - 1 | 1 )$.
Calculate the coordinates of $Q$ and show that the point $R ( 9 | - 1 | 4 )$ lies on the sphere.

Justify without further calculation that the triangle $P Q R$ is right-angled at $R$.

Given is the sphere $K$ with center $M ( 3 | - 6 | 5 )$ and radius $2 \sqrt { 6 }$.
Give an equation of $K$ in coordinate form and show that the point $P ( 5 | - 4 | 1 )$ lies on $K$.

Investigate whether $K$ intersects the $x _ { 1 } x _ { 2 }$-plane.

At the point that corresponds to point C in the model, the skier's line of travel is continued without a kink by a circular arc curve. During travel along this curve, the skier reaches a point that corresponds to point $D ( 18 | - 2 | 2 )$. The circular arc that describes this curve is part of a circle with center $M \left( m _ { 1 } \left| m _ { 2 } \right| m _ { 3 } \right)$. The coordinates of $M$ can be determined using the following system of equations. I $\quad m _ { 1 } + m _ { 2 } + 2 m _ { 3 } - 20 = 0$ II $\quad \left( \begin{array} { l } m _ { 1 } - 9 \\ m _ { 2 } - 1 \\ m _ { 3 } - 5 \end{array} \right) \circ \left( \begin{array} { c } 1.8 \\ 0.2 \\ - 1 \end{array} \right) = 0$ III $\sqrt { \left( m _ { 1 } - 9 \right) ^ { 2 } + \left( m _ { 2 } - 1 \right) ^ { 2 } + \left( m _ { 3 } - 5 \right) ^ { 2 } } = \sqrt { \left( m _ { 1 } - 18 \right) ^ { 2 } + \left( m _ { 2 } + 2 \right) ^ { 2 } + \left( m _ { 3 } - 2 \right) ^ { 2 } }$ Explain the geometric considerations that underlie equations I, II and III.

6. Determine the equation of the spherical surface $S$, with centre on the line $r: \left\{ \begin{array}{l} x = t \\ y = t \\ z = t \end{array} \right. t \in \mathbb{R}$ tangent to the plane $\pi: 3x - y - 2z + 14 = 0$ at the point $T(-4, 0, 1)$.

2. Consider the spherical surface with equation $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + z ^ { 2 } = 1$ and the plane $\pi$ with equation $x - 2 y - 2 z + d = 0$. Discuss, as the real parameter $d$ varies, whether the plane $\pi$ is secant, tangent or external to the spherical surface. Determine the value of the parameter $d$ so that $\pi$ divides the sphere into two equal parts.

If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are
(1) $( 4,9 , - 3 )$
(2) $( 4 , - 3,3 )$
(3) $( 4,3,5 )$
(4) $( 4,3 , - 3 )$

6. In coordinate space, $O$ is the origin and point $A$ has coordinates $(1, 2, 1)$. Let $S$ be the sphere with center $O$ and radius 4. What is the figure formed by all points $P$ on $S$ that satisfy the dot product $\overrightarrow{OA} \cdot \overrightarrow{OP} = 6$?
(1) Empty set
(2) A single point
(3) Two points
(4) A circle
(5) Two circles

In a spatial coordinate system, there is a globe with center at $O ( 0,0,0 )$ and north pole at $N ( 0,0,2 )$. A point $A$ on the sphere has coordinates $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } , \sqrt { 3 } \right)$. The point on the equator farthest from point $A$ is point $P$. On the great circle passing through points $A$ and $P$, the length of the minor arc between these two points is (blank). (Express as a fraction in lowest terms)

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) $\frac{4}{3}$