In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1.
a. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 0. b. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to $(-1)$. c. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 1. d. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 2.

In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
Let $\mathscr { D } _ { 1 }$ be the line with direction vector $\vec { u } ( 2 ; - 1 ; 1 )$ passing through A.
A parametric representation of the line $\mathscr { D } _ { 1 }$ is : a. $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 - t \\ z = 1 - t \end{array} \quad ( t \in \mathbb { R } ) \right.$ b. $\left\{ \begin{array} { l } x = - 1 + 2 t \\ y = 1 - t \\ z = 1 + t \end{array} \quad ( t \in \mathbb { R } ) \right.$ c. $\left\{ \begin{array} { l } x = 5 + 4 t \\ y = - 3 - 2 t \\ z = 1 + 2 t \end{array} \quad ( t \in \mathbb { R } ) \right.$ d. $\left\{ \begin{array} { l } x = 4 - 2 t \\ y = - 2 + t \\ z = 3 - 4 t \end{array} \quad ( t \in \mathbb { R } ) \right.$

In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
A measure of the angle $\widehat { \mathrm { BAC } }$ rounded to the nearest tenth of a degree is equal to : a. $22,2 ^ { \circ }$ b. $0,4 ^ { \circ }$ c. $67,8 ^ { \circ }$ d. $1,2 ^ { \circ }$

(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 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.

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 referred to an orthonormal coordinate system, consider the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ which have the following parametric representations respectively: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R} \quad \text{and} \quad \left\{ \begin{array}{l} x = -5t' + 3 \\ y = 2t' \\ z = t' + 4 \end{array} \right., t' \in \mathbb{R}$$
Statement 3: The lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant.
Indicate whether this statement is true or false, justifying your answer.

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?

Exercise 3 (5 points)

We place ourselves in space equipped with an orthonormal coordinate system whose origin is point A. We consider the points $\mathrm{B}(10; -8; 2)$, $\mathrm{C}(-1; -8; 5)$ and $\mathrm{D}(14; 4; 8)$.

1. a. Determine a system of parametric equations for each of the lines (AB) and (CD). b. Verify that the lines (AB) and (CD) are not coplanar.
2. We consider the point I on the line (AB) with abscissa 5 and the point J on the line (CD) with abscissa 4. a. Determine the coordinates of points I and J and deduce the distance IJ. b. Demonstrate that the line (IJ) is perpendicular to the lines (AB) and (CD). The line (IJ) is called the common perpendicular to the lines (AB) and (CD).
3. The purpose of this question is to verify that the distance IJ is the minimum distance between the lines (AB) and (CD). We consider a point $M$ on the line (AB) distinct from point I. We consider a point $M'$ on the line (CD) distinct from point J. a. Justify that the parallel to the line (IJ) passing through point $M'$ intersects the line $\Delta$ (the line parallel to (CD) passing through I) at a point that we will denote $P$. b. Demonstrate that the triangle $MPM'$ is right-angled at $P$. c. Justify that $MM' > IJ$ and conclude.

Exercise 3

Common to all candidates

Two species of turtles endemic to a small island in the Pacific Ocean, green turtles and hawksbill turtles, meet during different breeding episodes on two of the island's beaches to lay eggs. This island, being the convergence point of many turtles, specialists decided to take advantage of this to collect various data on them. They first observed that the corridors used in the ocean by each of the two species to reach the island could be assimilated to rectilinear trajectories. In what follows, space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 100 meters. The plane ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ) represents the water level and we admit that a point $M ( x ; y ; z )$ with $z < 0$ is located in the ocean. The specialists' model establishes that:

• the trajectory used in the ocean by green turtles is supported by the line $\mathscr { D } _ { 1 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 3 + t \\ y & = 6 t \text { with } t \text { real; } \\ z & = - 3 t \end{aligned} \right.$$

• the trajectory used in the ocean by hawksbill turtles is supported by the line $\mathscr { D } _ { 2 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 10 k \\ y & = 2 + 6 k \text { with } k \text { real; } \\ z & = - 4 k \end{aligned} \right.$$

1. Prove that the two species are never likely to cross before arriving on the island.
2. The objective of this question is to estimate the minimum distance separating these two trajectories. a. Verify that the vector $\vec { n } \left( \begin{array} { c } 3 \\ 13 \\ 27 \end{array} \right)$ is normal to the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$. b. It is admitted that the minimum distance between the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$ is the distance $\mathrm { HH } ^ { \prime }$ where $\overrightarrow { \mathrm { HH } ^ { \prime } }$ is a vector collinear to $\vec { n }$ with H belonging to the line $\mathscr { D } _ { 1 }$ and $\mathrm { H } ^ { \prime }$ belonging to the line $\mathscr { D } _ { 2 }$. Determine an approximate value in meters of this minimum distance. One may use the results below provided by a computer algebra system

\multicolumn{2}{|l|}{$\triangleright$ Computer algebra}
1	\begin{tabular}{ l } Solve $( \{ 10 * k - 3 - t = 3 * l , 2 + 6 * k - 6 * t = 13 * l , - 4 * k + 3 * t = 27 * l \} , \{ k , l , t \} )$
$\rightarrow \left\{ \left\{ k = \frac { 675 } { 1814 } , \ell = \frac { 17 } { 907 } , t = \frac { 603 } { 907 } \right\} \right\}$

\hline \end{tabular}

1. The scientists decide to install a beacon at sea.

It is located at point B with coordinates ( $2 ; 4 ; 0$ ). a. Let $M$ be a point on the line $\mathscr { D } _ { 1 }$.
Determine the coordinates of the point $M$ such that the distance $\mathrm { B} M$ is minimal. b. Deduce the minimum distance, rounded to the nearest meter, between the beacon and the green turtles.

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?

The objective of this exercise is to study the trajectories of two submarines in the diving phase. We consider that these submarines move in a straight line, each at constant speed. At each instant $t$, expressed in minutes, the first submarine is located by the point $S_{1}(t)$ and the second submarine is located by the point $S_{2}(t)$ in an orthonormal reference frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit metre. The plane defined by $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ represents the sea surface. The $z$ coordinate is zero at sea level, negative underwater.

1. We admit that, for every real $t \geqslant 0$, the point $S_{1}(t)$ has coordinates: $$\left\{ \begin{array}{l} x(t) = 140 - 60t \\ y(t) = 105 - 90t \\ z(t) = -170 - 30t \end{array} \right.$$ a. Give the coordinates of the submarine at the beginning of the observation. b. What is the speed of the submarine? c. We place ourselves in the vertical plane containing the trajectory of the first submarine.
   Determine the angle $\alpha$ that the submarine's trajectory makes with the horizontal plane. Give the value of $\alpha$ rounded to 0.1 degree.
2. At the beginning of the observation, the second submarine is located at the point $S_{2}(0)$ with coordinates $(68; 135; -68)$ and reaches after three minutes the point $S_{2}(3)$ with coordinates $(-202; -405; -248)$ at constant speed. At what instant $t$, expressed in minutes, are the two submarines at the same depth?

This exercise is a multiple choice questionnaire. For each question, four answers are proposed and only one of them is correct. No justification is required. 1.5 points are awarded for each correct answer. No points are deducted for no answer or an incorrect answer.
Question 1 In space with an orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$), we consider the line ($D$) with parametric representation $\left\{\begin{array}{l} x = 2 + t \\ y = 1 - 3t \\ z = 2t \end{array} \quad (t \in \mathbb{R})\right.$, and the plane $(P)$ with Cartesian equation $x + y + z - 3 = 0$.
We can assert that: Answer A: the line ($D$) and the plane ($P$) are strictly parallel. Answer B: the line ($D$) is contained in the plane ($P$). Answer C: the line ($D$) and the plane ($P$) intersect at the point with coordinates $(4; -5; 4)$. Answer D: the line ($D$) and the plane ($P$) are orthogonal.
Question 2 In the computer department of a large store, only one salesperson is present and there are many customers. We assume that the random variable $T$, which associates to each customer the waiting time in minutes for the salesperson to be available, follows an exponential distribution with parameter $\lambda$. The average waiting time is 20 minutes. Given that a customer has already waited 20 minutes, the probability that their total waiting time exceeds half an hour is: Answer A: $\mathrm{e}^{-\frac{1}{2}}$ Answer B: $\mathrm{e}^{-\frac{3}{2}}$ Answer C: $1 - \mathrm{e}^{-\frac{1}{2}}$ Answer D: $1 - \mathrm{e}^{-10\lambda}$
Question 3
A factory manufactures tennis balls in large quantities. To comply with international competition regulations, the diameter of a ball must be between $63.5\text{ mm}$ and $66.7\text{ mm}$. We denote by $D$ the random variable which associates to each ball produced its diameter measured in millimetres. We assume that $D$ follows a normal distribution with mean 65.1 and standard deviation $\sigma$. We call $P$ the probability that a ball chosen at random from the total production is compliant. The factory decides to adjust the machines so that $P$ equals 0.99. The value of $\sigma$, rounded to the nearest hundredth, allowing this objective to be achieved is: Answer A: 0.69 Answer B: 2.58 Answer C: 0.62 Answer D: 0.80
Question 4 The curve below is the graph, in an orthonormal coordinate system, of the function $f$ defined by: $$f(x) = \frac{4x}{x^{2} + 1}$$ The exact value of the positive real number $a$ such that the line with equation $x = a$ divides the shaded region into two regions of equal area is: Answer A: $\sqrt{\sqrt{\frac{3}{2}}}$ Answer B: $\sqrt{\sqrt{5} - 1}$ Answer C: $\ln 5 - 0.5$ Answer D: $\frac{10}{9}$

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.

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.

For each of the following questions, only one of the four statements is correct. Indicate on your answer sheet the question number and copy the letter corresponding to the correct statement. One point is awarded if the letter corresponds to the correct statement, 0 otherwise.
Throughout the exercise, we work in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ in space. The four questions are independent. No justification is required.

1. Consider the plane $P$ with Cartesian equation $3x + 2y + 9z - 5 = 0$ and the line $d$ with parametric representation: $\left\{\begin{array}{l} x = 4t + 3 \\ y = -t + 2 \\ z = -t + 9 \end{array}, t \in \mathbb{R}\right.$. Statement A: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(3;2;9)$. Statement B: plane $P$ and line $d$ are orthogonal. Statement C: plane $P$ and line $d$ are parallel. Statement D: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(-353; 91; 98)$.
   
2. Consider the cube ABCDEFGH and the points I, J and K defined by the vector equalities: $$\overrightarrow{\mathrm{AI}} = \frac{3}{4}\overrightarrow{\mathrm{AB}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{HK}} = \frac{3}{4}\overrightarrow{\mathrm{HG}}$$ Statement A: the cross-section of cube ABCDEFGH by plane (IJK) is a triangle. Statement B: the cross-section of cube ABCDEFGH by plane (IJK) is a quadrilateral. Statement C: the cross-section of cube ABCDEFGH by plane (IJK) is a pentagon. Statement D: the cross-section of cube ABCDEFGH by plane (IJK) is a hexagon.
   
3. Consider the line $d$ with parametric representation $\left\{\begin{aligned} x &= t + 2 \\ y &= 2 \\ z &= 5t - 6 \end{aligned}\right.$, with $t \in \mathbb{R}$, and the point $\mathrm{A}(-2; 1; 0)$. Let $M$ be a variable point on line $d$. Statement A: the smallest length $AM$ is equal to $\sqrt{53}$. Statement B: the smallest length $AM$ is equal to $\sqrt{27}$. Statement C: the smallest length $AM$ is attained when point $M$ has coordinates $(-2; 1; 0)$. Statement D: the smallest length $AM$ is attained when point $M$ has coordinates $(2; 2; -6)$.
   
4. Consider the plane $P$ with Cartesian equation $x + 2y - 3z + 1 = 0$ and the plane $P'$ with Cartesian equation $2x - y + 2 = 0$. Statement A: planes $P$ and $P'$ are parallel. Statement B: the intersection of planes $P$ and $P'$ is a line passing through points $\mathrm{A}(5; 12; 10)$ and $\mathrm{B}(3; 1; 2)$. Statement C: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{C}(2; 6; 5)$ and having a direction vector $\vec{u}(1; 2; 2)$. Statement D: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{D}(-1; 0; 0)$ and having a direction vector $\vec{v}(3; 6; 5)$.

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.

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?

Exercise 4 (Candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points $\mathrm { I } , \mathrm { J } , \mathrm { K } , \mathrm { L } , \mathrm { M }$ and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$ in which, for example, the point N has coordinates $\left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; 1 \right)$.

1. a. Give the coordinates of the vectors $\overrightarrow { \mathrm { NC } }$ and $\overrightarrow { \mathrm { ML } }$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.

Exercise 4 — Candidates who have NOT followed the specialisation course
For each of the following statements, indicate whether it is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. No answer is not penalised.
1. Statement 1: The equation $( 3 \ln x - 5 ) \left( e ^ { x } + 4 \right) = 0$ has exactly two real solutions.
2. Consider the sequence ( $u _ { n }$ ) defined by $$u _ { 0 } = 2 \text { and, for all natural number } n , u _ { n + 1 } = 2 u _ { n } - 5 n + 6 \text {. }$$ Statement 2: For all natural number $n , u _ { n } = 3 \times 2 ^ { n } + 5 n - 1$.
3. Consider the sequence ( $u _ { n }$ ) defined, for all natural number $n$, by $u _ { n } = n ^ { 2 } + \frac { 1 } { 2 }$.
Statement 3: The sequence $\left( u _ { n } \right)$ is geometric.
4. In a coordinate system of space, let $d$ be the line passing through point $\mathrm { A } ( - 3 ; 7 ; - 12 )$ and with direction vector $\vec { u } ( 1 ; - 2 ; 5 )$.
Let $d ^ { \prime }$ be the line with parametric representation $\left\{ \begin{array} { r l } x & = 2 t - 1 \\ y & = - 4 t + 3 \\ z & = 10 t - 2 . \end{array} , t \in \mathbf { R } \right.$
Statement 4: The lines $d$ and $d ^ { \prime }$ are coincident.
5. Consider a cube $A B C D E F G H$. The space is equipped with the orthonormal coordinate system ( $A$; $\overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
A parametric representation of the line (AG) is $\left\{ \begin{array} { l } x = t \\ y = t \\ z = t \end{array} \quad t \in \mathbf { R } \right.$.
Consider a point $M$ on the line (AG).
Statement 5: There are exactly two positions of point $M$ on the line (AG) such that the lines $( M \mathrm { ~B} )$ and $( M \mathrm { D } )$ are orthogonal.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer or the absence of an answer to a question earns or loses no points.
Space is referred to an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:

• The line $\mathscr{D}$ passing through the points $\mathrm{A}(1;1;-2)$ and $\mathrm{B}(-1;3;2)$.
• The line $\mathscr{D}'$ with parametric representation: $\left\{ \begin{aligned} x &= -4 + 3t \\ y &= 6 - 3t \\ z &= 8 - 6t \end{aligned} \right.$ with $t \in \mathbb{R}$.
• The plane $\mathscr{P}$ with Cartesian equation $x + my - 2z + 8 = 0$ where $m$ is a real number.

Question 1: Among the following points, which one belongs to the line $\mathscr{D}'$? a. $\mathrm{M}_1(-1;3;-2)$ b. $\mathrm{M}_2(11;-9;-22)$ c. $\mathrm{M}_3(-7;9;2)$ d. $\mathrm{M}_4(-2;3;4)$
Question 2: A direction vector of the line $\mathscr{D}'$ is: a. $\overrightarrow{u_1}\left(\begin{array}{c}-4\\6\\8\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}3\\3\\6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}3\\-3\\-6\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{c}-1\\3\\2\end{array}\right)$
Question 3: The lines $\mathscr{D}$ and $\mathscr{D}'$ are: a. intersecting b. strictly parallel c. non-coplanar d. coincident
Question 4: The value of the real number $m$ for which the line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$ is: a. $m = -1$ b. $m = 1$ c. $m = 5$ d. $m = -2$

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question earns neither points nor deducts points. The four questions are independent.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.

1. Consider the points $A(1; 0; 3)$ and $B(4; 1; 0)$.
   A parametric representation of the line (AB) is: a. $\left\{ \begin{aligned} x & = 3 + t \\ y & = 1 \\ z & = -3 + 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ b. $\left\{ \begin{array}{l} x = 1 + 4t \\ y = 3 \\ z = 3 \end{array} \right.$ with $t \in \mathbb{R}$ c. $\left\{ \begin{aligned} x & = 1 + 3t \\ y & = t \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ d. $\left\{ \begin{aligned} x & = 4 + t \\ y & = 1 \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$
   
2. Consider the line (d) with parametric representation $\left\{ \begin{aligned} x & = 3 + 4t \\ y & = 6t \\ z & = 4 - 2t \end{aligned} \right.$ with $t \in \mathbb{R}$
   Among the following points, which one belongs to the line (d)? a. $M(7; 6; 6)$ b. $N(3; 6; 4)$ c. $P(4; 6; -2)$ d. $R(-3; -9; 7)$
   
3. Consider the line $(d')$ with parametric representation $\left\{ \begin{aligned} x & = -2 + 3k \\ y & = -1 - 2k \\ z & = 1 + k \end{aligned} \right.$ with $k \in \mathbb{R}$
   The lines $(d)$ and $(d')$ are: a. secant b. non-coplanar c. parallel d. coincident
   
4. Consider the plane $(P)$ passing through the point $I(2; 1; 0)$ and perpendicular to the line (d).
   An equation of the plane $(P)$ is: a. $2x + 3y - z - 7 = 0$ b. $-x + y - 4z + 1 = 0$ c. $4x + 6y - 2z + 9 = 0$ d. $2x + y + 1 = 0$

A passage of an aerial acrobatics show in a duo is modelled as follows:

• we place ourselves in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath } , \vec { k })$, where one unit represents one metre;
• plane no. 1 must travel from point O to point $A(0 ; 200 ; 0)$ along a straight trajectory, at the constant speed of $200 \mathrm {~m/s}$;
• plane no. 2 must travel from point $B(-33 ; 75 ; 44)$ to point $C(87 ; 75 ; -116)$ also along a straight trajectory, and at the constant speed of $200 \mathrm {~m/s}$;
• at the same instant, plane no. 1 is at point O and plane no. 2 is at point B.

1. Justify that plane no. 2 will take the same time to travel segment $[BC]$ as plane no. 1 to travel segment $[OA]$.
2. Show that the trajectories of the two planes intersect.
3. Is there a risk of collision between the two planes during this passage?