Two lines $\ell _ { 1 }$ and $\ell _ { 2 }$ in 3-dimensional space are given by $$\ell _ { 1 } = \{ ( t - 9 , - t + 7 , 6 ) \mid t \in \mathbb { R } \} \quad \text{and} \quad \ell _ { 2 } = \{ ( 7 , s + 3 , 3 s + 4 ) \mid s \in \mathbb { R } \}.$$
Questions
(31) The plane passing through the origin and not intersecting either of $\ell _ { 1 }$ and $\ell _ { 2 }$ has equation $ax + by + cz = d$. Write the value of $| a + b + c + d |$ where $a, b, c, d$ are integers with $\gcd = 1$. (32) Let $r$ be the smallest possible radius of a circle that has a point on $\ell _ { 1 }$ as well as a point on $\ell _ { 2 }$. It is given that $r ^ { 2 }$ (i.e., the square of the smallest radius) is an integer. Write the value of $r ^ { 2 }$.

Let $\alpha$ be the plane passing through point $\mathrm { A } ( 1,2,3 )$ and perpendicular to the line $l : x - 1 = \frac { y - 2 } { - 2 } = \frac { z - 3 } { 3 }$. When the intersection point of plane $\alpha$ and line $m : x - 2 = y = \frac { z - 6 } { 5 }$ is B, what is the length of segment AB? [3 points]
(1) $\sqrt { 19 }$
(2) $\sqrt { 17 }$
(3) $\sqrt { 15 }$
(4) $\sqrt { 13 }$
(5) $\sqrt { 11 }$

As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]
(1) $\frac { \sqrt { 10 } } { 5 }$
(2) $\frac { \sqrt { 10 } } { 10 }$
(3) $\frac { \sqrt { 11 } } { 11 }$
(4) $\frac { 2 \sqrt { 11 } } { 11 }$
(5) $\frac { 3 \sqrt { 11 } } { 11 }$

In coordinate space, there are two points $\mathrm { A } ( 3,1,1 ) , \mathrm { B } ( 1 , - 3 , - 1 )$. For a point P on the plane $x - y + z = 0$, what is the minimum value of $| \overrightarrow { \mathrm { PA } } + \overrightarrow { \mathrm { PB } } |$? [4 points]
(1) $\frac { 4 \sqrt { 3 } } { 3 }$
(2) $\frac { 5 \sqrt { 3 } } { 3 }$
(3) $2 \sqrt { 3 }$
(4) $\frac { 7 \sqrt { 3 } } { 3 }$
(5) $\frac { 8 \sqrt { 3 } } { 3 }$

A sphere with center $\mathrm { C } ( 0,1,1 )$ and radius $2 \sqrt { 2 }$ intersects the line $\frac { x } { 2 } = y = - z$ at two points A and B. Let $S$ be the area of triangle CAB. Find the value of $S ^ { 2 }$.

Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 points]

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, let $C$ be the circle formed by the intersection of the sphere $S : x^2 + y^2 + z^2 = 4$ and the plane $\alpha : y - \sqrt{3}z = 2$. For point $\mathrm{A}(0, 2, 0)$ on circle $C$, let $\mathrm{P}$ and $\mathrm{Q}$ be the endpoints of a diameter of circle $C$ such that $\overline{\mathrm{AP}} = \overline{\mathrm{AQ}}$. Let $\mathrm{R}$ be another point where the line passing through $\mathrm{P}$ and perpendicular to plane $\alpha$ meets sphere $S$. If the area of triangle $\mathrm{ARQ}$ is $s$, find the value of $s^2$. [4 points]

On plane $\alpha$, there is a right isosceles triangle ABC with $\angle \mathrm { A } = 90 ^ { \circ }$ and $\overline { \mathrm { BC } } = 6$. A point P outside plane $\alpha$ is at a distance of 4 from the plane, and the foot of the perpendicular from P to plane $\alpha$ is point A. What is the distance from point P to line BC? [3 points]
(1) $3 \sqrt { 2 }$
(2) 5
(3) $3 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$
(5) 6

In coordinate space, let A be the intersection point of the line $\frac { x } { 2 } = y = z + 3$ and the plane $\alpha : x + 2 y + 2 z = 6$. A sphere with center at point $( 1 , - 1,5 )$ passing through point A intersects plane $\alpha$ to form a figure with area $k \pi$. Find the value of $k$. [3 points]

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, 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) $3 \sqrt { 2 }$

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, a line $l : \frac { x } { 2 } = 6 - y = z - 6$ and plane $\alpha$ meet perpendicularly at point $\mathrm { P } ( 2,5,7 )$. For a point $\mathrm { A } ( a , b , c )$ on line $l$ and a point Q on plane $\alpha$, when $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } } = 6$, what is the value of $a + b + c$? (Here, $a > 0$) [4 points]
(1) 15
(2) 16
(3) 17
(4) 18
(5) 19

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) $\frac { 10 } { 3 } \pi$

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, 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) $\overline { \mathrm { BC } } < \overline { \mathrm { AC } } < \overline { \mathrm { AB } }$

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]