(For candidates who have not followed the specialization course)
In space equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the tetrahedron ABCD whose vertices have coordinates: $$\mathrm{A}(1;-\sqrt{3};0);\quad \mathrm{B}(1;\sqrt{3};0);\quad \mathrm{C}(-2;0;0);\quad \mathrm{D}(0;0;2\sqrt{2}).$$

1. Prove that the plane (ABD) has the Cartesian equation $4x + z\sqrt{2} = 4$.
2. We denote by $\mathscr{D}$ the line whose parametric representation is $$\left\{\begin{array}{l} x = t \\ y = 0 \\ z = t\sqrt{2} \end{array}, t \in \mathbb{R}\right.$$ a. Prove that $\mathscr{D}$ is the line that is parallel to $(\mathrm{CD})$ and passes through O. b. Determine the coordinates of point G, the intersection of the line $\mathscr{D}$ and the plane (ABD).
3. a. We denote by L the midpoint of segment $[\mathrm{AC}]$. Prove that the line (BL) passes through point O and is orthogonal to the line (AC). b. Prove that triangle ABC is equilateral and determine the centre of its circumscribed circle.
4. Prove that the tetrahedron ABCD is regular, that is, a tetrahedron whose six edges all have the same length.

Exercise 4 — Candidates who have not followed the specialization course
In space, we consider a tetrahedron ABCD whose faces ABC, ACD and ABD are right-angled and isosceles triangles at A. We denote by E, F and G the midpoints of sides $[\mathrm{AB}]$, $[\mathrm{BC}]$ and $[\mathrm{CA}]$ respectively. We choose AB as the unit of length and we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}})$ of space.

1. We denote by $\mathscr { P }$ the plane that passes through A and is perpendicular to the line $(\mathrm{DF})$. We denote by H the point of intersection of plane $\mathscr { P }$ and line (DF). a. Give the coordinates of points D and F. b. Give a parametric representation of line (DF). c. Determine a Cartesian equation of plane $\mathscr { P }$. d. Calculate the coordinates of point H. e. Prove that the angle $\widehat{\mathrm{EHG}}$ is a right angle.
2. We denote by $M$ a point on line (DF) and by $t$ the real number such that $\overrightarrow{\mathrm{DM}} = t \overrightarrow{\mathrm{DF}}$. We denote by $\alpha$ the measure in radians of the geometric angle $\widehat{\mathrm{EMG}}$. The purpose of this question is to determine the position of point $M$ so that $\alpha$ is maximum. a. Prove that $ME^{2} = \frac{3}{2} t^{2} - \frac{5}{2} t + \frac{5}{4}$. b. Prove that triangle $M\mathrm{EG}$ is isosceles at $M$. Deduce that $ME \sin\left(\frac{\alpha}{2}\right) = \frac{1}{2\sqrt{2}}$. c. Justify that $\alpha$ is maximum if and only if $\sin\left(\frac{\alpha}{2}\right)$ is maximum. Deduce that $\alpha$ is maximum if and only if $ME^{2}$ is minimum. d. Conclude.

In space, we consider a pyramid SABCE with square base ABCE with centre O. Let D be the point in space such that ( O ; $\overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OD } }$ ) is an orthonormal frame. The point S has coordinates $( 0 ; 0 ; 3 )$ in this frame.

Part A

1. Let U be the point on the line $( \mathrm { SB } )$ with height 1. Construct the point U on the attached figure in appendix 1.
2. Let V be the intersection point of the plane (AEU) and the line (SC). Show that the lines (UV) and (BC) are parallel. Construct the point V on the attached figure in appendix 1.
3. Let K be the point with coordinates $\left( \frac { 5 } { 6 } ; - \frac { 1 } { 6 } ; 0 \right)$. Show that K is the foot of the altitude from U in the trapezoid AUVE.

Part B

In this part, we admit that the area of the quadrilateral AUVE is $\frac { 5 \sqrt { 43 } } { 18 }$.

1. We admit that the point $U$ has coordinates $\left( 0 ; \frac { 2 } { 3 } ; 1 \right)$. Verify that the plane (EAU) has equation $3 x - 3 y + 5 z - 3 = 0$.
2. Give a parametric representation of the line (d) perpendicular to the plane (EAU) passing through the point $S$.
3. Determine the coordinates of H, the intersection point of the line (d) and the plane (EAU).
4. The plane (EAU) divides the pyramid (SABCE) into two solids. Do these two solids have the same volume?

In space equipped with an orthonormal coordinate system, we consider:

• the points $\mathrm { A } ( 0 ; 1 ; - 1 )$ and $\mathrm { B } ( - 2 ; 2 ; - 1 )$.
• the line $\mathscr { D }$ with parametric representation $\left\{ \begin{array} { r l } x & = - 2 + t \\ y & = 1 + t \\ z & = - 1 - t \end{array} , t \in \mathbb { R } \right.$.

1. Determine a parametric representation of the line ( AB ).
2. a. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not parallel. b. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not intersecting.

In the following, the letter $u$ denotes a real number. We consider the point $M$ of the line $\mathscr { D }$ with coordinates ( $- 2 + u ; 1 + u ; - 1 - u$ ).
3. Verify that the plane $\mathscr { P }$ with equation $x + y - z - 3 u = 0$ is orthogonal to the line $\mathscr { D }$ and passes through the point $M$.
4. Show that the plane $\mathscr { P }$ and the line (AB) intersect at a point $N$ with coordinates $( - 4 + 6 u ; 3 - 3 u ; - 1 )$.
5. a. Show that the line $( M N )$ is perpendicular to the line $\mathscr { D }$. b. Does there exist a value of the real number $u$ for which the line ( $M N$ ) is perpendicular to the line $( \mathrm { AB } )$ ? 6. a. Express $M N ^ { 2 }$ as a function of $u$. b. Deduce the value of the real number $u$ for which the distance $M N$ is minimal.

An artist wishes to create a sculpture composed of a tetrahedron placed on a cube with 6-metre edges. These two solids are represented by the cube $ABCDEFGH$ and by the tetrahedron $SELM$.
The space is equipped with an orthonormal coordinate system $(A; \overrightarrow{AI}, \overrightarrow{AJ}, \overrightarrow{AK})$ such that: $I \in [AB]$, $J \in [AD]$, $K \in [AE]$ and $AI = AJ = AK = 1$, the graphical unit representing 1 metre.
The points $L$, $M$ and $S$ are defined as follows:

• $L$ is the point such that $\overrightarrow{FL} = \frac{2}{3}\overrightarrow{FE}$;
• $M$ is the point of intersection of the plane $(BDL)$ and the line $(EH)$;
• $S$ is the point of intersection of the lines $(BL)$ and $(AK)$.

1. Prove, without calculating coordinates, that the lines $(LM)$ and $(BD)$ are parallel.
2. Prove that the coordinates of point $L$ are $(2; 0; 6)$.
3. a. Give a parametric representation of the line $(BL)$. b. Verify that the coordinates of point $S$ are $(0; 0; 9)$.
4. Let $\vec{n}$ be the vector with coordinates $(3; 3; 2)$. a. Verify that $\vec{n}$ is normal to the plane $(BDL)$. b. Prove that a Cartesian equation of the plane $(BDL)$ is: $$3x + 3y + 2z - 18 = 0$$ c. It is admitted that the line $(EH)$ has the parametric representation: $$\left\{\begin{array}{l} x = 0 \\ y = s \\ z = 6 \end{array} \quad (s \in \mathbb{R})\right.$$ Calculate the coordinates of point $M$.
5. Calculate the volume of the tetrahedron $SELM$. Recall that the volume $V$ of a tetrahedron is given by the following formula: $$V = \frac{1}{3} \times \text{Area of base} \times \text{Height}$$
6. The artist wishes the measure of angle $\widehat{SLE}$ to be between $55^\circ$ and $60^\circ$. Is this angle constraint satisfied?

We place ourselves in an orthonormal coordinate system with origin O and axes $( \mathrm { O } x )$, $( \mathrm { O } y )$ and $( \mathrm { O } z )$. In this coordinate system, we are given the points $\mathrm { A } ( - 3 ; 0 ; 0 ) , \mathrm { B } ( 3 ; 0 ; 0 ) , \mathrm { C } ( 0 ; 3 \sqrt { 3 } ; 0 )$ and $\mathrm { D } ( 0 ; \sqrt { 3 } ; 2 \sqrt { 6 } )$. We denote H as the midpoint of segment [CD] and I as the midpoint of segment [BC].

1. Calculate the lengths AB and AD.

We admit for the rest that all edges of the solid ABCD have the same length, that is, the tetrahedron ABCD is a regular tetrahedron. We call $\mathscr { P }$ the plane with normal vector $\overrightarrow { \mathrm { OH } }$ and passing through point I.
2. Study of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$ a. Show that a Cartesian equation of plane $\mathscr { P }$ is : $2 y \sqrt { 3 } + z \sqrt { 6 } - 9 = 0$. b. Prove that the midpoint J of $[ \mathrm { BD } ]$ is the intersection point of line (BD) and plane $\mathscr { P }$. c. Give a parametric representation of line (AD), then prove that plane $\mathscr { P }$ and line (AD) intersect at a point K whose coordinates you will determine. d. Prove that lines (IJ) and (JK) are perpendicular. e. Determine precisely the nature of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$.
3. Can we place a point M on edge $[ \mathrm { BD } ]$ such that triangle OIM is right-angled at M?

Let ABCDEFGH be a cube. We consider:

• I and J the midpoints respectively of segments [AD] and [BC];
• P the center of face ABFE, that is, the intersection of diagonals (AF) and (BE);
• Q the midpoint of segment [FG].

We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AE } } )$. Throughout the exercise, we may use the coordinates of the points in the figure without justifying them. We admit that a parametric representation of the line (IJ) is
$$\left\{ \begin{array} { l } x = r \\ y = 1 , \quad r \in \mathbf { R } \\ z = 0 \end{array} \right.$$

1. Verify that a parametric representation of the line (PQ) is

$$\left\{ \begin{array} { l } x = 1 + t \\ y = t , \quad t \in \mathbf { R } \\ z = 1 + t \end{array} \right.$$
Let $t$ be a real number and $\mathrm { M } ( 1 + t ; t ; 1 + t )$ be the point on the line (PQ) with parameter $t$.
2. a. We admit that there exists a unique point K belonging to the line (IJ) such that (MK) is orthogonal to (IJ). Prove that the coordinates of this point $\mathrm { K }$ are $( 1 + t ; 1 ; 0 )$. b. Deduce that $\mathrm { MK } = \sqrt { 2 + 2 t ^ { 2 } }$.
3. a. Verify that $y - z = 0$ is a Cartesian equation of the plane (HGB). b. We admit that there exists a unique point L belonging to the plane (HGB) such that (ML) is orthogonal to (HGB). Verify that the coordinates of this point L are $\left( 1 + t ; \frac { 1 } { 2 } + t ; \frac { 1 } { 2 } + t \right)$. c. Deduce that the distance ML is independent of $t$.
4. Does there exist a value of $t$ for which the distance MK is equal to the distance ML?

Exercise 3 (5 points)
The purpose of this exercise is to examine, in different cases, whether the altitudes of a tetrahedron are concurrent, that is, to study the existence of an intersection point of its four altitudes. We recall that in a tetrahedron MNPQ, the altitude from M is the line passing through M perpendicular to the plane (NPQ).
Part A: Study of particular cases
We consider a cube ABCDEFGH. We admit that the lines (AG), (BH), (CE) and (DF), called ``main diagonals'' of the cube, are concurrent.

1. We consider the tetrahedron ABCE. a. Specify the altitude from E and the altitude from C in this tetrahedron. b. Are the four altitudes of the tetrahedron ABCE concurrent?
   
2. We consider the tetrahedron ACHF and work in the coordinate system $(\mathrm{A} ; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. a. Verify that a Cartesian equation of the plane (ACH) is: $x - y + z = 0$. b. Deduce that (FD) is the altitude from F of the tetrahedron ACHF. c. By analogy with the previous result, specify the altitudes of the tetrahedron ACHF from the vertices $\mathrm{A}$, $\mathrm{C}$ and H respectively. Are the four altitudes of the tetrahedron ACHF concurrent?

In the rest of this exercise, a tetrahedron whose four altitudes are concurrent will be called an orthocentric tetrahedron.
Part B: A property of orthocentric tetrahedra
In this part, we consider a tetrahedron MNPQ whose altitudes from vertices M and N intersect at a point K. The lines (MK) and (NK) are therefore perpendicular to the planes (NPQ) and (MPQ) respectively.

1. a. Justify that the line (PQ) is perpendicular to the line (MK); we admit likewise that the lines (PQ) and (NK) are perpendicular. b. What can we deduce from the previous question regarding the line (PQ) and the plane (MNK)? Justify the answer.
2. Show that the edges [MN] and [PQ] are perpendicular.

Thus, we obtain the following property: If a tetrahedron is orthocentric, then its opposite edges are perpendicular in pairs.
Part C: Application
In an orthonormal coordinate system, we consider the points: $$\mathrm { R } ( - 3 ; 5 ; 2 ) , \mathrm { S } ( 1 ; 4 ; - 2 ) , \mathrm { T } ( 4 ; - 1 ; 5 ) \quad \text { and } \mathrm { U } ( 4 ; 7 ; 3 ) .$$ Is the tetrahedron RSTU orthocentric? Justify.

We consider a cube $ABCDEFGH$. The point M is the midpoint of $[\mathrm{BF}]$, I is the midpoint of [BC], the point N is defined by the relation $\overrightarrow{\mathrm{CN}} = \frac{1}{2}\overrightarrow{\mathrm{GC}}$ and the point P is the center of the face ADHE.

Part A:

1. Justify that the line (MN) intersects the segment [BC] at its midpoint I.
2. Construct, on the figure provided in the appendix, the cross-section of the cube by the plane (MNP).

Part B:

We equip space with the orthonormal coordinate system ($A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

1. Justify that the vector $\vec{n}\left(\begin{array}{l}1\\2\\2\end{array}\right)$ is a normal vector to the plane (MNP).
   Deduce a Cartesian equation of the plane (MNP).
2. Determine a system of parametric equations of the line (d) passing through G and orthogonal to the plane (MNP).
3. Show that the line (d) intersects the plane (MNP) at the point K with coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{1}{3}\right)$. Deduce the distance GK.
4. We admit that the four points M, E, D and I are coplanar and that the area of the quadrilateral MEDI is $\frac{9}{8}$ square units. Calculate the volume of the pyramid GMEDI.

Exercise 2 -- Common to all candidates
Alex and Élisa, two drone pilots, are training on a terrain consisting of a flat part bordered by an obstacle. We consider an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), with one unit corresponding to ten metres. Six points are defined by their coordinates: $$\mathrm { O } ( 0 ; 0 ; 0 ) , \mathrm { P } ( 0 ; 10 ; 0 ) , \mathrm { Q } ( 0 ; 11 ; 1 ) , \mathrm { T } ( 10 ; 11 ; 1 ) , \mathrm { U } ( 10 ; 10 ; 0 ) \text { and } \mathrm { V } ( 10 ; 0 ; 0 )$$ The flat part is delimited by the rectangle OPUV and the obstacle by the rectangle PQTU.
The two drones are assimilable to two points and follow rectilinear trajectories:

• Alex's drone follows the trajectory carried by the line $( \mathrm { AB } )$ with $\mathrm { A } ( 2 ; 4 ; 0.25 )$ and $\mathrm { B } ( 2 ; 6 ; 0.75 )$;
• Élisa's drone follows the trajectory carried by the line (CD) with C(4; 6; 0.25) and D(2; 6; 0.25).

Part A: Study of Alex's drone trajectory

1. Determine a parametric representation of the line ( AB ).
2. 1. [a.] Justify that the vector $\vec { n } ( 0 ; 1 ; - 1 )$ is a normal vector to the plane (PQU).
   2. [b.] Deduce a Cartesian equation of the plane (PQU).
3. Prove that the line (AB) and the plane (PQU) intersect at the point I with coordinates $\left( 2 ; \frac { 37 } { 3 } ; \frac { 7 } { 3 } \right)$.
4. Explain why, following this trajectory, Alex's drone does not encounter the obstacle.

Part B: Minimum distance between the two trajectories
To avoid a collision between their two devices, Alex and Élisa impose a minimum distance of 4 metres between the trajectories of their drones. For this, we consider a point $M$ on the line (AB) and a point $N$ on the line (CD). There then exist two real numbers $a$ and $b$ such that $\overrightarrow { \mathrm { A } M } = a \overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { C } N } = b \overrightarrow { \mathrm { CD } }$.

1. Prove that the coordinates of the vector $\overrightarrow { M N }$ are $( 2 - 2 b ; 2 - 2 a ; - 0.5 a )$.
2. It is admitted that the lines (AB) and (CD) are not coplanar. It is also admitted that the distance $MN$ is minimal when the line ( $MN$ ) is perpendicular to both the line ( AB ) and the line (CD). Prove then that the distance $MN$ is minimal when $a = \frac { 16 } { 17 }$ and $b = 1$.
3. Deduce the minimum value of the distance $MN$ and conclude.

Let ABCDEFGH be a cube and I the center of the square ADHE, that is, the midpoint of segment [AH] and segment [ED]. Let J be a point on segment [CG]. The cross-section of the cube ABCDEFGH by the plane (FIJ) is the quadrilateral FKLJ.
We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. We have therefore $\mathrm{A}(0;0;0)$, $\mathrm{B}(1;0;0)$, $\mathrm{D}(0;1;0)$ and $\mathrm{E}(0;0;1)$. Parts A and B can be treated independently.

Part A

In this part, the point J has coordinates $\left(1; 1; \frac{2}{5}\right)$.

1. Prove that the coordinates of point I are $\left(0; \frac{1}{2}; \frac{1}{2}\right)$.
2. a. Prove that the vector $\vec{n}\begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}$ is a normal vector to the plane (FIJ). b. Prove that a Cartesian equation of the plane (FIJ) is $$-x + 3y + 5z - 4 = 0.$$
3. Let $d$ be the line perpendicular to the plane (FIJ) and passing through B. a. Determine a parametric representation of the line $d$. b. We denote by M the point of intersection of the line $d$ and the plane (FIJ). Prove that $\mathrm{M}\left(\frac{6}{7}; \frac{3}{7}; \frac{5}{7}\right)$.
4. a. Calculate $\overrightarrow{\mathrm{BM}} \cdot \overrightarrow{\mathrm{BF}}$. b. Deduce an approximate value to the nearest degree of the angle $\widehat{\mathrm{MBF}}$.

Part B

In this part, J is an arbitrary point on segment [CG]. Its coordinates are therefore $(1; 1; a)$, where $a$ is a real number in the interval $[0; 1]$.

1. Show that the cross-section of the cube by the plane (FIJ) is a parallelogram.
2. We admit that L has coordinates $\left(0; 1; \frac{a}{2}\right)$. For which value(s) of $a$ is the quadrilateral FKLJ a rhombus?

As shown in the figure for question (20), in the triangular pyramid $\mathrm { P } - \mathrm { ABC }$, plane $\mathrm { PAC } \perp$ plane $\mathrm { ABC }$, $\angle \mathrm { ABC } = \frac { \pi } { 2 }$. Points $D$ and $E$ lie on segment $AC$ with $\mathrm { AD } = \mathrm { DE } = \mathrm { EC } = 2$, $\mathrm { PD } = \mathrm { PC } = 4$. Point $F$ lies on segment $AB$ with $\mathrm { EF } \parallel \mathrm { BC }$ .
(I) Prove that $\mathrm { AB } \perp$ plane $PFE$ .
(II) If the volume of the quadrangular pyramid $\mathrm { P } - \mathrm { DFBC }$ is 7, find the length of segment $BC$.

20. In ``The Nine Chapters on the Mathematical Art,'' a quadrangular pyramid with a rectangular base and one lateral edge perpendicular to the base is called a ``yang ma'', and a tetrahedron with all four faces being right triangles is called a ``bie nao''. In the yang ma $\mathrm { P } - \mathrm { ABCD }$ shown in the figure, the lateral edge $\mathrm { PD } \perp$ base ABCD, and $\mathrm { PD } = \mathrm { CD }$. Point E is the midpoint of PC. Connect $\mathrm { DE } , \mathrm { BD } , \mathrm { BE }$.
[Figure]
Figure for Question 20
(I) Prove that $\mathrm { DE } \perp$ plane PBC. Determine whether the tetrahedron EBCD is a ``bie nao''. If yes, write out the right angle of each face (only conclusions are needed); if no, please explain the reason; (II) Let the volume of the yang ma $\mathr

As shown in the figure, in rectangular prism $\mathrm { ABCD } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, we have $\mathrm { AB } = 16 , \mathrm { BC } = 10 , \mathrm { AA } _ { 1 } = 8$. Points $\mathrm { E }$ and $\mathrm { F }$ are on $\mathrm { A } _ { 1 } \mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 } \mathrm { C } _ { 1 }$ respectively, with $\mathrm { A } _ { 1 } \mathrm { E } = \mathrm { D } _ { 1 } \mathrm { F }$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square.
(I) Draw this square in the figure (no need to explain the method or reasoning)
(II) Find the sine of the angle between line

18. (This question is worth 12 points)

A net of a cube and a schematic diagram of the cube are shown in the figure. (1) Mark the letters $F , G , H$ at the corresponding vertices of the cube (no explanation needed); (2) Determine the positional relationship between plane $B E G$ and plane $A C H$, and prove your conclusion; (3) Prove: line $D F \perp$ plane $B E G$. [Figure] [Figure]

18. A net of a cube and a schematic diagram of the cube are shown in the figure. In the cube, let $M$ be the midpoint of $BC$ and $N$ be the midpoint of $GH$.
(1) Mark the letters $F$, $G$, $H$ at the corresponding vertices of the cube (no explanation needed).
(2) Prove: line $MN \parallel$ plane $BDH$.
(3) Find the cosine of the dihedral angle $A - E G - M$. [Figure]

17. (13 points) As shown in the figure, $AA _ { 1 } \perp$ plane $ABC$, $BB _ { 1 } \parallel AA _ { 1 }$, $AB = AC = 3$, $BC = 2 \sqrt { 5 }$, $AA _ { 1 } = \sqrt { 7 }$, $BB _ { 1 } = 2 \sqrt { 7 }$. Points $E$ and $F$ are the midpoints of $BC$ and $A _ { 1 } C$ respectively. (I) Prove that $EF \parallel$ plane $A _ { 1 } B _ { 1 } BA$; (II) Prove that plane $AEA _ { 1 } \perp$ plane $BCB _ { 1 }$. (III) Find the angle between line $A _ { 1 } B _ { 1 }$ and plane $BCB _ { 1 }$. [Figure]

As shown in the figure, in the quadrangular prism $\mathrm{ABCD} - A_1B_1C_1D_1$, the lateral edge $AA_1 \perp$ base $\mathrm{ABCD}$, $\mathrm{AB} \perp \mathrm{AC}$, $\mathrm{AB} = 1$, $\mathrm{AC} = AA_1 = 2$, $AD = CD = \sqrt{5}$, and points M and N are the midpoints of $B_1C$ and $D_1D$ respectively.
(I) Prove: $\mathrm{MN} \parallel$ plane ABCD
(II) Find the sine value of the dihedral angle $D_1 - AC - B_1$;
(III) Let E be a point on edge $A_1B_1$. If the sine value of the angle between line NE and plane ABCD is $\frac{1}{3}$, find the length of segment $A_1E$.

18. (15 points) As shown in the figure, in the triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ , $\angle \mathrm { ABC } = 90 ^ { \circ } , \mathrm { AB } = \mathrm { AC } = 2 , \mathrm { AA } _ { 1 } = 4$ , the projection of $A _ { 1 }$ on the base plane ABC is the midpoint of BC, and D is the midpoint of $B _ { 1 } C _ { 1 }$.
(1) Prove that $A _ { 1 } \mathrm { D } \perp$ plane $\mathrm { A } _ { 1 } \mathrm { BC }$ ;
(2) Find the sine of the angle between line $\mathrm { A } _ { 1 } \mathrm { B}$ and plane $\mathrm { BB } _ { 1 } \mathrm { C } C _ { 1 }$ . [Figure]

17. (This question is worth 15 points) As shown in the figure, in the triangular prism $ABC - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle BAC = 90 ^ { \circ }$ , $AB = AC = 2$ , $A _ { 1 } A = 4$ , the projection of $A _ { 1 }$ on the base plane $ABC$ is the midpoint of $BC$, and $D$ is the midpoint of $B _ { 1 } C _ { 1 }$ . (I) Prove that $A _ { 1 } D \perp$ plane $A _ { 1 } B C _ { 1 }$ ; (II) Find the cosine of the plane angle of the dihedral angle $A _ { 1 } - BD - B _ { 1 }$ . [Figure]

As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.

19. (12 points) Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2.
(1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ .
(2) Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ , equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ . Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ .
From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .

19. In the quadrangular pyramid $Q - A B C D$ , the base $A B C D$ is a square with $A D = 2$ , $Q D = Q A = \sqrt { 5 }$ , $Q C = 3$ . [Figure]
(1) Prove: plane $Q A D \perp$ plane $A B C D$ ;
(2) Find the cosine of the dihedral angle $B - Q D - A$ . Answer: (1) See proof below; (2) $\frac { 2 } { 3 }$ .

[Solution]

[Analysis] (1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . We can prove that $Q O \perp$ plane $A B C D$ , thus obtaining plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Establish a coordinate system as shown in the figure. After finding the normal vectors of planes $Q A D$ and $B Q D$ , we can find the cosine of the dihedral angle.

[Detailed Solution]

[Figure]
(1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . Since $Q A = Q D$ and $O A = O D$ , we have $Q O \perp A D$ . Since $A D = 2$ and $Q A = \sqrt { 5 }$ , we have $Q O = \sqrt { 5 - 1 } = 2$ . In square $A B C D$ , since $A D = 2$ , we have $D O = 1$ , thus $C O = \sqrt { 1 + 4 } = \sqrt { 5 }$ . Since $Q C = 3$ , we have $Q C ^ { 2 } = 9 = 4 + 5 = Q O ^ { 2 } + O C ^ { 2 }$ , so $\triangle Q O C$ is a right triangle with $Q O \perp O C$ . Since $O C \cap A D = O$ , we have $Q O \perp$ plane $A B C D$ . Since $Q O \subset$ plane $Q A D$ , we have plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Combined with $Q O \perp$ plane $A B C D$ from part (1), we can establish a coordinate system as shown in the figure. [Figure]
Then $D ( 0,1,0 ) , Q ( 0,0,2 ) , B ( 2 , -1,0 )$ , so $\overrightarrow { B Q } = ( -2,1,2 ) , \overrightarrow { B D } = ( -2,2,0 )$ . Let the normal vector of plane $Q B D$ be $\vec { n } = ( x , y , z )$ . Then $\left\{ \begin{array} { l } \vec { n } \cdot \overrightarrow { B Q } = 0 \\ \vec { n } \cdot \overrightarrow { B D } = 0 \end{array} \right.$ , i.e., $\left\{ \begin{array} { l } - 2 x + y + 2 z = 0 \\ - 2 x + 2 y = 0 \end{array} \right.$ . Taking $x = 1$ , we get $y = 1 , z = \frac { 1 } { 2 }$ , Thus $\vec { n } = \left( 1,1 , \frac { 1 } { 2 } \right)$ . The normal vector of plane $Q A D$ is $\vec { m } = ( 1,0,0 )$ . Therefore $\cos \langle \vec { m } , \vec { n } \rangle = \frac { |\vec{m} \cdot \vec{n}| } { |\vec{m}| \cdot |\vec{n}| } = \frac

Consider a pyramid $OPQRS$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP = 3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS = 3$. Then
(A) the acute angle between $OQ$ and $OS$ is $\frac{\pi}{3}$
(B) the equation of the plane containing the triangle $OQS$ is $x - y = 0$
(C) the length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
(D) the perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$

Let $P _ { 1 } : 2 x + y - z = 3$ and $P _ { 2 } : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of $P _ { 1 }$ and $P _ { 2 }$ has direction ratios $1,2 , - 1$
(B) The line $$\frac { 3 x - 4 } { 9 } = \frac { 1 - 3 y } { 9 } = \frac { z } { 3 }$$ is perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$
(C) The acute angle between $P _ { 1 }$ and $P _ { 2 }$ is $60 ^ { \circ }$
(D) If $P _ { 3 }$ is the plane passing through the point $( 4,2 , - 2 )$ and perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$, then the distance of the point $( 2,1,1 )$ from the plane $P _ { 3 }$ is $\frac { 2 } { \sqrt { 3 } }$