For a regular hexahedron (cube) $\mathrm { ABCD } - \mathrm { EFGH }$, let $\theta$ be the angle between plane AFG and plane AGH. What is the value of $\cos ^ { 2 } \theta$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$

In coordinate space, let B be the foot of the perpendicular from the point $\mathrm { A } ( 3,6,0 )$ to the plane $\sqrt { 3 } y - z = 0$. Find the value of $\overrightarrow { \mathrm { OA } } \cdot \overrightarrow { \mathrm { OB } }$. (Here, O is the origin.) [4 points]

In coordinate space, consider the triangle ABC with vertices $\mathrm { A } ( 54,0,0 ) , \mathrm { B } ( 0,27,0 ) , \mathrm { C } ( 0,0,27 )$ on the plane $x + 2 y + 2 z = 54$. A point $\mathrm { P } ( x , y , z )$ is in the interior of triangle ABC. Let Q be the orthogonal projection of P onto the $xy$-plane, R be the orthogonal projection of P onto the $yz$-plane, and S be the orthogonal projection of P onto the $zx$-plane. When $\overline { \mathrm { QR } } = \overline { \mathrm { QS } }$, find the maximum volume of the tetrahedron QPRS. [4 points]

As shown in the figure, on a plane $\alpha$ there is an equilateral triangle ABC with side length 3, and a sphere $S$ with radius 2 is tangent to the plane $\alpha$ at point A. For a point D on the sphere $S$ such that the segment AD passes through the center O of the sphere $S$, find the value of $| \overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { DC } } | ^ { 2 }$. [4 points]

In coordinate space, let $l$ be the line of intersection of the plane $x = 3$ and the plane $z = 1$. When point P moves on line $l$, what is the minimum value of the length of segment OP? (Here, O is the origin.) [3 points]
(1) $2 \sqrt { 2 }$
(2) $\sqrt { 10 }$
(3) $2 \sqrt { 3 }$
(4) $\sqrt { 14 }$
(5) $3 \sqrt { 2 }$

In coordinate space, the figure $S$ is formed by the intersection of the sphere $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 9$ with center C and the plane $x + y + z = 6$. For two points $\mathrm { P } , \mathrm { Q }$ on figure $S$, what is the minimum value of the dot product $\overrightarrow { \mathrm { CP } } \cdot \overrightarrow { \mathrm { CQ } }$ of the two vectors $\overrightarrow { \mathrm { CP } } , \overrightarrow { \mathrm { CQ } }$? [4 points]
(1) - 3
(2) - 2
(3) - 1
(4) 1
(5) 2

In coordinate space, there is a tetrahedron ABCD with vertices at four points $\mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( - 3,0,0 )$, $\mathrm { D } ( 0,0,2 )$. For point P moving on edge BD, let the coordinates of point P that minimize $\overline { \mathrm { PA } } ^ { 2 } + \overline { \mathrm { PC } } ^ { 2 }$ be $( a , b , c )$. If $a + b + c = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

In coordinate space, the distance between point $\mathrm { P } ( 0,3,0 )$ and point $\mathrm { A } ( - 1,1 , a )$ is 2 times the distance between point P and point $\mathrm { B } ( 1,2 , - 1 )$. What is the value of the positive number $a$? [2 points]
(1) $\sqrt { 7 }$
(2) $\sqrt { 6 }$
(3) $\sqrt { 5 }$
(4) 2
(5) $\sqrt { 3 }$

In coordinate space, triangle ABC satisfies the following conditions. (가) The area of triangle ABC is 6. (나) The area of the orthogonal projection of triangle ABC onto the $yz$-plane is 3.
What is the maximum area of the orthogonal projection of triangle ABC onto the plane $x - 2 y + 2 z = 1$? [4 points]
(1) $2 \sqrt { 6 } + 1$
(2) $2 \sqrt { 2 } + 3$
(3) $3 \sqrt { 5 } - 1$
(4) $2 \sqrt { 5 } + 1$
(5) $3 \sqrt { 6 } - 2$

In coordinate space, there is a point $\mathrm { A } ( 9,0,5 )$, and on the $xy$-plane there is an ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$. For a point P on the ellipse, find the maximum value of $\overline { \mathrm { AP } }$. [3 points]

In coordinate space, for two points $\mathrm { A } ( a , 1,3 ) , \mathrm { B } ( a + 6,4,12 )$, the point that divides the line segment AB internally in the ratio $1 : 2$ has coordinates $( 5,2 , b )$. What is the value of $a + b$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

In coordinate space, one face ABC of a regular tetrahedron ABCD lies on the plane $2 x - y + z = 4$, and the vertex D lies on the plane $x + y + z = 3$. When the centroid of triangle ABC has coordinates $( 1,1,3 )$, what is the length of one edge of the regular tetrahedron ABCD? [4 points]
(1) $2 \sqrt { 2 }$
(2) 3
(3) $2 \sqrt { 3 }$
(4) 4
(5) $3 \sqrt { 2 }$

As shown in the figure, there is a rectangular piece of paper ABCD with $\overline { \mathrm { AB } } = 9$ and $\overline { \mathrm { AD } } = 3$. Using the line connecting point E on segment AB and point F on segment DC as the fold line, the paper is folded so that the orthogonal projection of point B onto the plane AEFD is point D. When $\overline { \mathrm { AE } } = 3$, the angle between the two planes AEFD and EFCB is $\theta$. Find the value of $60 \cos \theta$. (Given that $0 < \theta < \frac { \pi } { 2 }$ and the thickness of the paper is negligible.) [4 points]

In coordinate space, for two points $\mathrm { A } ( a , 5,2 ) , \mathrm { B } ( - 2,0,7 )$, the point that divides segment AB internally in the ratio $3 : 2$ has coordinates $( 0 , b , 5 )$. What is the value of $a + b$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, when the line passing through two points $\mathrm { A } ( 5,5 , a ) , \mathrm { B } ( 0,0,3 )$ is perpendicular to the line $x = 4 - y = z - 1$, what is the value of $a$? [3 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

In coordinate space, there are two points $\mathrm { P } , \mathrm { Q }$ moving on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$. Let $\mathrm { P } _ { 1 } , \mathrm { Q } _ { 1 }$ be the feet of the perpendiculars from points P and Q to the plane $y = 4$ respectively, and let $\mathrm { P } _ { 2 } , \mathrm { Q } _ { 2 }$ be the feet of the perpendiculars to the plane $y + \sqrt { 3 } z + 8 = 0$ respectively. Find the maximum value of $2 | \overrightarrow { \mathrm { PQ } } | ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 1 } \mathrm { Q } _ { 1 } } \right| ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } } \right| ^ { 2 }$. [4 points]

In coordinate space, for two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 3 , b )$, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a + b$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

Let $l$ be the line passing through two distinct points $\mathrm { A } , \mathrm { B }$ on plane $\alpha$, and let H be the foot of the perpendicular from point P (not on plane $\alpha$) to plane $\alpha$. When $\overline { \mathrm { AB } } = \overline { \mathrm { PA } } = \overline { \mathrm { PB } } = 6 , \overline { \mathrm { PH } } = 4$, what is the distance between point H and line $l$? [3 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$

In coordinate space, there is a sphere $S : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 50$ and a point $\mathrm { P } ( 0,5,5 )$. For all circles $C$ satisfying the following conditions, find the maximum area of the orthogonal projection of $C$ onto the $xy$-plane, expressed as $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points] (가) Circle $C$ is formed by the intersection of a plane passing through point P and the sphere $S$. (나) The radius of circle $C$ is 1.

In coordinate space, for three points $\mathrm { A } ( a , 0,5 ) , \mathrm { B } ( 1 , b , - 3 ) , \mathrm { C } ( 1,1,1 )$ that are vertices of a triangle, when the centroid of the triangle has coordinates $( 2,2,1 )$, what is the value of $a + b$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

In coordinate space, there are a point $\mathrm { A } ( 2,2,1 )$ and a plane $\alpha : x + 2 y + 2 z - 14 = 0$. When point P on plane $\alpha$ satisfies $\overline { \mathrm { AP } } \leq 3$, what is the area of the projection of the figure traced by point P onto the $xy$-plane? [4 points]
(1) $\frac { 14 } { 3 } \pi$
(2) $\frac { 13 } { 3 } \pi$
(3) $4 \pi$
(4) $\frac { 11 } { 3 } \pi$
(5) $\frac { 10 } { 3 } \pi$

In coordinate space, there are two mutually perpendicular planes $\alpha$ and $\beta$. For two points $\mathrm { A }$ and $\mathrm { B }$ on plane $\alpha$, $\overline { \mathrm { AB } } = 3 \sqrt { 5 }$, and line AB is parallel to plane $\beta$. The distance between point A and plane $\beta$ is 2, and the distance between a point P on plane $\beta$ and plane $\alpha$ is 4. Find the area of triangle PAB. [4 points]

For two points $\mathrm { A } ( 2 , \sqrt { 2 } , \sqrt { 3 } )$ and $\mathrm { B } ( 1 , - \sqrt { 2 } , 2 \sqrt { 3 } )$ in coordinate space, point P satisfies the following conditions. (가) $| \overrightarrow { \mathrm { AP } } | = 1$ (나) The angle between $\overrightarrow { \mathrm { AP } }$ and $\overrightarrow { \mathrm { AB } }$ is $\frac { \pi } { 6 }$.
For point Q on a sphere centered at the origin with radius 1, the maximum value of $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } }$ is $a + b \sqrt { 33 }$. Find the value of $16 \left( a ^ { 2 } + b ^ { 2 } \right)$. (Here, $a$ and $b$ are rational numbers.) [4 points]

In coordinate space, for two points $\mathrm { A } ( 1 , a , - 6 ) , \mathrm { B } ( - 3,2 , b )$, when the point that externally divides the line segment AB in the ratio $3 : 2$ lies on the $x$-axis, what is the value of $a + b$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$

In coordinate space, let $\vec { a } , \vec { b } , \vec { c }$ be the position vectors of three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ with respect to the origin. The dot products between these vectors are shown in the following table.

$\cdot$	$\vec { a }$	$\vec { b }$	$\vec { c }$
$\vec { a }$	2	1	$- \sqrt { 2 }$
$\vec { b }$	1	2	0
$\vec { c }$	$- \sqrt { 2 }$	0	2

For example, $\vec { a } \cdot \vec { c } = - \sqrt { 2 }$. Which of the following correctly shows the order of the distances between the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$? [4 points]
(1) $\overline { \mathrm { AB } } < \overline { \mathrm { AC } } < \overline { \mathrm { BC } }$
(2) $\overline { \mathrm { AB } } < \overline { \mathrm { BC } } < \overline { \mathrm { AC } }$
(3) $\overline { \mathrm { AC } } < \overline { \mathrm { AB } } < \overline { \mathrm { BC } }$
(4) $\overline { \mathrm { BC } } < \overline { \mathrm { AB } } < \overline { \mathrm { AC } }$
(5) $\overline { \mathrm { BC } } < \overline { \mathrm { AC } } < \overline { \mathrm { AB } }$