In a regular tetrahedron ABCD with edge length 4, let O be the centroid of triangle ABC and P be the midpoint of segment AD. For a point Q on face BCD of the regular tetrahedron ABCD, when the two vectors $\overrightarrow { \mathrm { OQ } }$ and $\overrightarrow { \mathrm { OP } }$ are perpendicular to each other, the maximum value of $| \overrightarrow { \mathrm { PQ } } |$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

For two points $\mathrm { A } ( 1,6,4 ) , \mathrm { B } ( a , 2 , - 4 )$ in coordinate space, the point that divides segment AB internally in the ratio $1 : 3$ has coordinates $( 2,5,2 )$. What is the value of $a$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

In coordinate space, there are three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ not on the same line. For a plane $\alpha$ satisfying the following conditions, let $d ( \alpha )$ be the minimum distance among the distances from each point $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to plane $\alpha$.
(a) Plane $\alpha$ intersects segment AC and also intersects segment BC.
(b) Plane $\alpha$ does not intersect segment AB.
Among planes $\alpha$ satisfying the above conditions, let $\beta$ be the plane where $d ( \alpha )$ is maximized. Which of the following statements in are correct? [4 points]
$\text{ㄱ}$. Plane $\beta$ is perpendicular to the plane passing through the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$. $\text{ㄴ}$. Plane $\beta$ passes through the midpoint of segment AC or the midpoint of segment BC. $\text{ㄷ}$. When the three points are $\mathrm { A } ( 2,3,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( 2 , - 1,0 )$, $d ( \beta )$ equals the distance between point B and plane $\beta$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In coordinate space, there is a circle $C$ formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 6$ and the plane $x + 2 z - 5 = 0$. Let P be the point on circle $C$ with the minimum $y$-coordinate, and let Q be the foot of the perpendicular from point P to the $xy$-plane. For a point X moving on circle $C$, the maximum value of $| \overrightarrow { \mathrm { PX } } + \overrightarrow { \mathrm { QX } } | ^ { 2 }$ is $a + b \sqrt { 30 }$.
Find the value of $10 ( a + b )$. (Here, $a$ and $b$ are rational numbers.) [4 points]

For two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 2,1 )$ in coordinate space, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, the $x$-coordinate of the point where the plane passing through the point $( 2,0,5 )$ and containing the line $x - 1 = 2 - y = \frac { z + 1 } { 2 }$ meets the $x$-axis is? [3 points]
(1) $\frac { 9 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3
(5) $\frac { 5 } { 2 }$

For a tetrahedron ABCD with an equilateral triangle BCD of side length 12 as one face, let H be the foot of the perpendicular from vertex A to plane BCD. The point H lies inside triangle BCD. The area of triangle CDH is 3 times the area of triangle BCH, the area of triangle DBH is 2 times the area of triangle BCH, and $\overline { \mathrm { AH } } = 3$. Let M be the midpoint of segment BD, and let Q be the foot of the perpendicular from point A to segment CM. What is the length of segment AQ? [4 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$

For two points $\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 3,2,0 )$ in coordinate space, if the coordinates of a point on the $y$-axis that is equidistant from both points is $( 0 , a , 0 )$, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

As shown in the figure, there is a rhombus-shaped piece of paper ABCD with side length 4 and $\angle \mathrm { BAD } = \frac { \pi } { 3 }$. Let M and N be the midpoints of sides BC and CD respectively. The paper is folded along the three line segments $\mathrm { AM } , \mathrm { AN } , \mathrm { MN }$ to form a tetrahedron PAMN. The area of the orthogonal projection of triangle AMN onto the plane PAM is $\frac { q } { p } \sqrt { 3 }$. Find the value of $p + q$. (Here, the thickness of the paper is neglected, P is the point where the three points $\mathrm { B } , \mathrm { C } , \mathrm { D }$ coincide when the paper is folded, and $p$ and $q$ are coprime natural numbers.) [4 points]

In coordinate space, for two points $\mathrm { A } ( 3 , - 3,3 ) , \mathrm { B } ( - 2,7 , - 2 )$, let $\alpha , \beta$ be the two planes that contain segment AB and are tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$. Let C and D be the points of tangency of the two planes $\alpha , \beta$ with the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ respectively. If the volume of tetrahedron ABCD is $\frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

In coordinate space, let P be the point obtained by reflecting point $\mathrm { A } ( 2,1,3 )$ across the xy-plane, and let Q be the point obtained by reflecting point A across the yz-plane. What is the length of segment PQ? [2 points]
(1) $5 \sqrt { 2 }$
(2) $2 \sqrt { 13 }$
(3) $3 \sqrt { 6 }$
(4) $2 \sqrt { 14 }$
(5) $2 \sqrt { 15 }$

As shown in the figure, there is a cube $\mathrm { ABCD } - \mathrm { EFGH }$ with edge length 4. Let M be the midpoint of segment AD. What is the area of triangle MEG? [3 points]
(1) $\frac { 21 } { 2 }$
(2) 11
(3) $\frac { 23 } { 2 }$
(4) 12
(5) $\frac { 25 } { 2 }$

In coordinate space, there is a sphere $$S : ( x - 2 ) ^ { 2 } + ( y - \sqrt { 5 } ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 25$$ with center $\mathrm { C } ( 2 , \sqrt { 5 } , 5 )$ passing through point $\mathrm { P } ( 0,0,1 )$. For a point Q moving on the circle formed by the intersection of sphere $S$ and plane OPC, and a point R moving on sphere $S$, let $\mathrm { Q } _ { 1 }$ and $\mathrm { R } _ { 1 }$ be the orthogonal projections of points $\mathrm { Q }$ and $\mathrm { R }$ onto the xy-plane respectively.
For two points $\mathrm { Q } , \mathrm { R }$ that maximize the area of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$, the area of the orthogonal projection of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$ onto plane PQR is $\frac { q } { p } \sqrt { 6 }$. Find the value of $p + q$. [4 points]

For two points $\mathrm{A}(a, -2, 6)$ and $\mathrm{B}(9, 2, b)$ in coordinate space, the midpoint of segment AB has coordinates $(4, 0, 7)$. What is the value of $a + b$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

There is a plane $\alpha$ in coordinate space. Let $\mathrm{A}'$ and $\mathrm{B}'$ be the orthogonal projections of two distinct points $\mathrm{A}$ and $\mathrm{B}$ (not on plane $\alpha$) onto plane $\alpha$, respectively. $$\overline{\mathrm{AB}} = \overline{\mathrm{A'B'}} = 6$$ Let $\mathrm{M}'$ be the orthogonal projection of the midpoint M of segment AB onto plane $\alpha$. A point P is chosen on plane $\alpha$ such that $$\overline{\mathrm{PM'}} \perp \overline{\mathrm{A'B'}}, \quad \overline{\mathrm{PM'}} = 6$$ When the area of the orthogonal projection of triangle $\mathrm{A'B'P}$ onto plane ABP is $\frac{9}{2}$, what is the length of segment PM? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

As shown in the figure, for a tetrahedron ABCD with $\overline{\mathrm{AB}} = 6$, $\overline{\mathrm{BC}} = 4\sqrt{5}$, let M be the midpoint of segment BC. Triangle AMD is equilateral and line BC is perpendicular to plane AMD. Find the area of the orthogonal projection of the circle inscribed in triangle ACD onto plane BCD. [3 points]
(1) $\frac{\sqrt{10}}{4}\pi$
(2) $\frac{\sqrt{10}}{6}\pi$
(3) $\frac{\sqrt{10}}{8}\pi$
(4) $\frac{\sqrt{10}}{10}\pi$
(5) $\frac{\sqrt{10}}{12}\pi$

In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points]
(1) $\sqrt{43}$
(2) $\sqrt{47}$
(3) $\sqrt{51}$
(4) $\sqrt{55}$
(5) $\sqrt{59}$

16. The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 2 t \\ y = 2 - t \end{array} ( t \in \mathbb{R} ) \right.$. Then a direction vector $\overrightarrow { Therefore $\frac { ( 1 - \cos \theta ) ^ { 2 } } { 4 } + \frac { ( 1 + \sin \theta ) ^ { 2 } } { 4 } < 1$ , simplifying gives $\sin \theta - \cos \theta < \frac { 1 } { 2 } , \sin \left( \theta - \frac { \pi } { 4 } \right) < \frac { \sqrt { 2 } } { 4 }$ , Also $0 < \theta < \pi$ , that is $- \frac { \pi } { 4 } < \theta - \frac { \pi } { 4 } < \frac { 3 \pi } { 4 }$ , so $- \frac { \pi } { 4 } < \theta - \frac { \pi } { 4 } < \arcsin \frac { \sqrt { 2 } } { 4 }$ , Thus the range of values for $\theta$ is $\left( 0 , \frac { \pi } { 4 } + \arcsin \frac { \sqrt { 2 } } { 4 } \right)$ .

20. (Total: 14 points; Part 1: 7 points; Part 2: 7 points) $ABCD - A_1B_1C_1D_1$ is a right square prism with base edge length 1 and height $AA_1 = 2$. Find:
(1) The angle between skew lines $BD$ and $AB_1$ (express the result using inverse trigonometric functions);
(2) The volume of tetrahedron $AB_1D_1C$ [Figure]

13. As shown in the figure, in the triangular pyramid $A - B C D$, $AB = AC = BD = CD = 3$ , $AD = BC = 2$ , and $M , N$ are the midpoints of $AD , BC$ respectively. Then the cosine of the angle between the skew lines $AN$ and $CM$ is $\_\_\_\_$ .

14. As shown in the figure, quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes they lie in are mutually perpendicular. A moving point $M$ is on segment $PQ$. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { AB }$ and $\mathrm { BC }$ respectively. Let the angle between skew lines $EM$ and $AF$ be $\theta$, then the maximum value of $\cos \theta$ is $\_\_\_\_$. [Figure]

15. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in space with $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the space vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = 2 , \vec { b } \cdot \vec { e } _ { 2 } = \frac { 5 } { 2 }$ , and for all $x , y \in \mathbb{R}$ , $\left| \vec { b } - \left( x \vec { e } _ { 1 } + y \vec { e } _ { 2 } \right) \right| \geq \left| \vec { b } - \left( x _ { 0 } \vec { e } _ { 1 } + y _ { 0 } \vec { e } _ { 2 } \right) \right| = 1$ ( $x _ { 0 } , y _ { 0 } \in \mathbb{R}$ ), then $x _ { 0 } =$ $\_\_\_\_$ , $y _ { 0 } =$ $\_\_\_\_$ , $| \vec { b } | =$ $\_\_\_\_$ . III. Solution Questions: This section contains 5 questions, for a total of 74 points. Solutions should include explanations, proofs, or calculation steps.

16. As shown in the figure, in the right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, given $A C \perp B C$. Let D be the midpoint of $A B _ { 1 }$, $B _ { 1 } C \cap B C _ { 1 } = E$ . Prove: (1) $D E \parallel$ plane $A A _ { 1 } C C _ { 1 }$
(2) $B C _ { 1 } \perp A B _ { 1 }$ [Figure]

17. (This question is worth 14 points) As shown in the figure, in the quadrangular pyramid $A - E F C B$, $\triangle A E F$ is an equilateral triangle, plane $A E F \perp$ plane $E F C B$, $E F / / B C$, $B C = 4$, $E F = 2 a$, $\angle E B C = \angle F C B = 60 ^ { \circ }$, and $O$ is the midpoint of $E F$. (I) Prove: $A O \perp B E$; (II) Find the cosine of the dihedral angle $F - A E - B$; (III) If $B E \perp$ plane $A O C$, find the value of $a$. [Figure]

17. In the geometric solid ABCDE shown in the figure, quadrilateral ABCD is a rectangle, $AB \perp$ plane $BEC$, $BE \perp EC$, $AB = BE = EC = 2$, and $G$ and $F$ are the midpoints of segments $BE$ and $DC$ respectively.
(1) Prove that $GF \parallel$ plane $ADE$.
(2) Find the cosine of the acute dihedral angle between plane $AEF$ and plane $BEC$.