In space, there is a unit cube with edge length 1. Point $O$ is one vertex, and the remaining 7 vertices are $A, B, C, D, E, F, G$. Given that $\overline { O A } = \overline { A B } = \overline { B C } = \overline { C D } = \overline { D E } = \overline { E F } = \overline { F G } = 1$ and $\overline { O G } > 1$, select the vertex farthest from point $O$.
(1) $C$
(2) $D$
(3) $E$
(4) $F$
(5) $G$

In coordinate space, there are three mutually perpendicular vectors $\vec { u } , \vec { v } , \vec { w }$. Given that $\vec { u } - \vec { v } = ( 2 , - 1,0 )$ and $\vec { v } - \vec { w } = ( - 1,2,3 )$. What is the volume of the parallelepiped spanned by $\vec { u } , \vec { v } , \vec { w }$?
(1) $2 \sqrt { 5 }$
(2) $5 \sqrt { 2 }$
(3) $2 \sqrt { 10 }$
(4) $4 \sqrt { 5 }$
(5) $4 \sqrt { 10 }$

In coordinate space, a plane intersects the plane $x = 0$ and the plane $z = 0$ at lines $L_{1}$ and $L_{2}$, respectively.
Given that $L_{1}$ and $L_{2}$ are parallel, $L_{1}$ passes through the point $(0, 2, -11)$, and $L_{2}$ passes through the point $(8, 21, 0)$,
the distance between $L_{1}$ and $L_{2}$ is $\sqrt{(10-1)(10-2)(10-3)}$. (Express as a simplified radical)

A corner of a classroom is formed by two walls and the floor, which are mutually perpendicular. Let the corner be point $O$. There is a triangular baffle $A B C$ with vertices $A , B , C$ located on the intersection lines between walls or between walls and the floor, at distances of 20, 20, and 10 centimeters from corner $O$ respectively. The three sides $\overline { A B }$ , $\overline { B C }$ , $\overline { C A }$ are flush with the walls or floor, as shown in the figure. Find the area of the triangular baffle $ABC$.

Problem 4

For the real numbers $\theta$ and $\alpha$ within the regions $0 \leq \theta < 2 \pi$ and $0 \leq \alpha \leq \pi$, consider the line L that passes through two points: point $\mathrm { P } ( \cos \theta$, $\sin \theta , 1 )$ and point $\mathrm { Q } ( \cos ( \theta + \alpha ) , \sin ( \theta + \alpha ) , - 1 )$ in a three-dimensional Cartesian coordinate system $x y z$. I. Represent the line L as a linear function of a parameter $t$. Here, the point on the line L at $t = 0$ should represent the point Q and the point at $t = 1$ should represent the point P. II. Find the surface S swept by the line L as an equation of $x , y$ and $z$ when $\theta$ varies in the region $0 \leq \theta < 2 \pi$. Let C be the intersection lines of the surface S with the plane $y = 0$. Find the equation of C in terms of $x$ and $z$, and sketch the shape of C.
Next, examine the Gaussian curvature of the surface S. Generally, when the position vector $r$ of a point R on a curved surface is represented using parameters $u$ and $v$ by
$$\boldsymbol { r } ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) ,$$
the Gaussian curvature $K$ is represented as the following equation:
$$K = \frac { \left( \boldsymbol { r } _ { u u } \cdot \boldsymbol{e} \right) \left( \boldsymbol { r } _ { v v } \cdot \boldsymbol { e } \right) - \left( \boldsymbol { r } _ { u v } \cdot \boldsymbol { e } \right) ^ { 2 } } { \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { u } \right) \left( \boldsymbol { r } _ { v } \cdot \boldsymbol { r } _ { v } \right) - \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { v } \right) ^ { 2 } } ,$$
where $\boldsymbol { r } _ { u }$ and $\boldsymbol { r } _ { v }$ are first-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$, and $\boldsymbol { r } _ { u u } , \boldsymbol { r } _ { u v }$ and $\boldsymbol { r } _ { v v }$ are second-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$. $( \boldsymbol { a } \cdot \boldsymbol { b } )$ represents the inner product of two three-dimensional vectors $a$ and $b$, and $e$ is the unit vector of the normal direction at the point R. III. Let the point W be the intersection of the surface S and the $x$ axis in the region $x > 0$. Calculate the Gaussian curvature of S at the point W for $\alpha$ within the region $0 \leq \alpha < \pi$. IV. For $\alpha$ within the region $0 \leq \alpha < \pi$, prove that the Gaussian curvature is less than or equal to 0 at arbitrary points on the surface $S$.

In the three-dimensional orthogonal coordinate system $x y z$, the unit vectors along the $x , y$, and $z$ directions are $\mathbf { i } , \mathbf { j }$, and $\mathbf { k }$, respectively. Using the parameter $\theta ( 0 \leq \theta \leq \pi )$, we define two curves by their vector functions $\mathbf { P } ( \theta )$ and $\mathbf { Q } ( \theta )$ :
$$\begin{aligned} & \mathbf { P } ( \theta ) = x ( \theta ) \mathbf { i } + y ( \theta ) \mathbf { j } \\ & \mathbf { Q } ( \theta ) = \mathbf { P } ( \theta ) + z ( \theta ) \mathbf { k } \end{aligned}$$
where
$$\begin{aligned} & x ( \theta ) = \frac { 3 } { 2 } \cos ( \theta ) - \frac { 1 } { 2 } \cos ( 3 \theta ) \\ & y ( \theta ) = \frac { 3 } { 2 } \sin ( \theta ) - \frac { 1 } { 2 } \sin ( 3 \theta ) \end{aligned}$$
Here, $z ( \theta )$ is a continuous function satisfying $z ( 0 ) > 0$ and $z ( \pi ) < 0$, and the curve parametrized by $\mathbf { Q } ( \theta )$ lies on the sphere of radius 2, centered at the origin $( 0,0,0 )$ of the coordinate system. The positive direction of a curve corresponds to increasing values of the parameter $\theta$. Note that the curvature is the reciprocal of the radius of curvature. Answer the following questions.
I. As $\theta$ is varied from 0 to $\pi$, calculate the arc length of the curve parametrized by $\mathbf { P } ( \theta )$.
II. Obtain $z ( \theta )$.
III. Let $\alpha$ be the angle between the tangent of the curve parametrized by $\mathbf { Q } ( \theta )$ and the unit vector $\mathbf { k }$. Calculate $\cos ( \alpha )$.
IV. Find the curvature $\kappa _ { P } ( \theta )$ of the curve parametrized by $\mathbf { P } ( \theta )$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.
V. Let $\kappa _ { Q } ( \theta )$ be the curvature of the curve parametrized by $\mathbf { Q } ( \theta )$. Express $\kappa _ { Q } ( \theta )$ in terms of $\kappa _ { P } ( \theta )$ and $\alpha$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.

5

In coordinate space, consider the circle of radius $1$ centered at the origin in the $xy$-plane. Let $S$ be the cone (including its interior) with this circle as its base and with vertex at the point $(0,\,0,\,2)$. Also, let $A(1,\,0,\,2)$.

1. [(1)] When point $P$ moves over the base of $S$, let $T$ be the region swept out by the line segment $AP$. Illustrate in the same plane the cross-section of $S$ by the plane $z=1$ and the cross-section of $T$ by the plane $z=1$.
   
2. [(2)] When point $P$ moves throughout $S$, find the volume of the region swept out by the line segment $AP$.

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4 (See the solution/explanation page)
Consider the four points $\mathrm{O}(0,\ 0,\ 0)$, $\mathrm{A}(2,\ 0,\ 0)$, $\mathrm{B}(1,\ 1,\ 1)$, $\mathrm{C}(1,\ 2,\ 3)$ in coordinate space.
(1) Find the coordinates of the point P satisfying $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OA}}$, $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OB}}$, $\overrightarrow{\mathrm{OP}} \cdot \overrightarrow{\mathrm{OC}} = 1$.
(2) Drop a perpendicular from point P to line AB, and let H be the intersection of that perpendicular with line AB. Express $\overrightarrow{\mathrm{OH}}$ in terms of $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$.
(3) Define point Q by $\overrightarrow{\mathrm{OQ}} = \dfrac{3}{4}\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OP}}$, and consider the sphere $S$ centered at Q with radius $r$.
Find the range of $r$ such that $S$ has a common point with triangle OHB. Here, triangle OHB lies in the plane containing the three points O, H, B, and consists of the boundary and its interior.
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Below is shown a structure made with two identical rectangular prisms with edge lengths of 2, 3, and 4 units. These prisms are placed adjacent to each other as shown in the figure.
According to this, what is the length of the line segment AB connecting vertices A and B in units?
A) $6 \sqrt { 2 }$
B) $8 \sqrt { 3 }$
C) $5 \sqrt { 5 }$
D) 7
E) 9