A municipality decides to replace the traditional July 14 fireworks with a luminous drone show. For drone piloting, space is equipped with an orthonormal reference frame $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$ whose unit is one hundred meters.
The position of each drone is modeled by a point and each drone is sent from a starting point D with coordinates $(2;5;1)$. It is desired to form figures with drones by positioning them in the same plane $\mathscr{P}$. Three drones are positioned at points $\mathrm{A}(-1;-1;17)$, $\mathrm{B}(4;-2;4)$ and $\mathrm{C}(1;-3;7)$.

1. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
   In the following, we denote by $\mathscr{P}$ the plane (ABC) and we consider the vector $\vec{n}\begin{pmatrix}2\\-3\\1\end{pmatrix}$.
   
2. a. Justify that $\vec{n}$ is normal to the plane (ABC). b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $2x - 3y + z - 18 = 0$.
   
3. The drone pilot decides to send a fourth drone taking as trajectory the line $d$ whose parametric representation is given by $$d : \left\{\begin{array}{rl} x &= 3t + 2 \\ y &= t + 5 \\ z &= 4t + 1 \end{array},\text{ with } t \in \mathbb{R}.\right.$$ a. Determine a direction vector of the line $d$. b. So that this new drone is also placed in the plane $\mathscr{P}$, determine by calculation the coordinates of point E, the intersection of the line $d$ with the plane $\mathscr{P}$.
   
4. The drone pilot decides to send a fifth drone along the line $\Delta$ which passes through point $\mathrm{D}$ and which is perpendicular to the plane $\mathscr{P}$. This fifth drone is also placed in the plane $\mathscr{P}$, at the intersection between the line $\Delta$ and the plane $\mathscr{P}$. We admit that the point $\mathrm{F}(6;-1;3)$ corresponds to this location. Prove that the distance between the starting point D and the plane $\mathscr{P}$ equals $2\sqrt{14}$ hundreds of meters.
   
5. The show organizer asks the pilot to send a new drone in the plane (no matter its position in the plane), always starting from point D. Knowing that there are 40 seconds left before the start of the show and that the drone flies in a straight trajectory at $18.6\,\mathrm{m.s}^{-1}$, can the new drone arrive on time?

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?

Space is referred to an orthonormal coordinate system $( O ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.\nWe consider:

• $\alpha$ any real number;
• the points $A ( 1 ; 1 ; 0 ) , B ( 2 ; 1 ; 0 )$ and $C ( \alpha ; 3 ; \alpha )$;
• (d) the line with parametric representation: $\left\{ \begin{array} { l } x = 1 + t \\ y = 2 t \\ z = - t \end{array} \quad , t \in \mathbf { R } \right.$.

For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.

Statement 1:

For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { \jmath } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$.

Statement 2:

There exists exactly one value of the real number $\alpha$ such that the lines $( A C )$ and $d$ are parallel.

Statement 3 :

A measure of the angle $\widehat { O A B }$ is $135 ^ { \circ }$.

Statement 4 :

The orthogonal projection of point $A$ onto the line $( d )$ is the point $H$ with coordinates: $H ( 1 ; 2 ; 2 )$.

Statement 5 :

The sphere with center $O$ and radius 1 meets the line ( $d$ ) at two distinct points.\nWe recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the line $(d)$ whose parametric representation is: $$\left\{\begin{array}{rl} x & = 3 - 2t \\ y & = -1 \\ z & = 2 - 6t \end{array}, \text{ where } t \in \mathbb{R}\right.$$ We also consider the following points:

• $\mathrm{A}(3; -3; -2)$
• $\mathrm{B}(5; -4; -1)$
• C the point on line $(d)$ with x-coordinate 2
• H the orthogonal projection of point B onto the plane $\mathscr{P}$ with equation $x + 3z - 7 = 0$

Statement 1: The line $(d)$ and the y-axis are two non-coplanar lines.
Statement 2: The plane passing through $A$ and perpendicular to line $(d)$ has the Cartesian equation: $$x + 3z + 3 = 0$$
Statement 3: A measure, expressed in radians, of the geometric angle $\widehat{\mathrm{BAC}}$ is $\frac{\pi}{6}$.
Statement 4: The distance BH is equal to $\frac{\sqrt{10}}{2}$.

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{C}(3; 0; 0)$, $\mathrm{D}(0; 2; 0)$, $\mathrm{H}(-6; 2; 2)$ and $\mathrm{J}\left(\frac{-54}{13}; \frac{62}{13}; 0\right)$;
• the plane $P$ with Cartesian equation $2x + 3y + 6z - 6 = 0$;
• the plane $P'$ with Cartesian equation $x - 2y + 3z - 3 = 0$;
• the line $(d)$ with a parametric representation: $\left\{\begin{array}{l} x = -8 + \frac{1}{3}t \\ y = -1 + \frac{1}{2}t \\ z = -4 + t \end{array}, t \in \mathbb{R}\right.$

For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.
Statement 1: The line $(d)$ is orthogonal to the plane $P$ and intersects this plane at $H$.
Statement 2: The measure in degrees of the angle $\widehat{\mathrm{DCH}}$, rounded to $10^{-1}$, is $17.3^{\circ}$.
Statement 3: The planes $P$ and $P'$ are secant and their intersection is the line $\Delta$ with a parametric representation: $\left\{\begin{array}{l} x = 3 - 3t \\ y = 0 \\ z = t \end{array}, t \in \mathbb{R}\right.$.
Statement 4: Point J is the orthogonal projection of point H onto the line (CD).

Space is referred to an orthonormal coordinate system $( O ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider:

• the points $C ( 3 ; 0 ; 0 )$, $D ( 0 ; 2 ; 0 )$, $H ( - 6 ; 2 ; 2 )$ and $J \left( \frac { - 54 } { 13 } ; \frac { 62 } { 13 } ; 0 \right)$;
• the plane $P$ with Cartesian equation $2 x + 3 y + 6 z - 6 = 0$;
• the plane $P ^ { \prime }$ with Cartesian equation $x - 2 y + 3 z - 3 = 0$;
• the line $( d )$ with a parametric representation: $\left\{ \begin{array} { l } x = - 8 + \frac { 1 } { 3 } t \\ y = - 1 + \frac { 1 } { 2 } t \\ z = - 4 + t \end{array} , t \in \mathbf { R } \right.$

For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.
Statement 1: The line $( d )$ is orthogonal to the plane $P$ and intersects this plane at $H$.
Statement 2: The measure in degrees of the angle $\widehat { D C H }$, rounded to $10 ^ { - 1 }$, is $17.3 ^ { \circ }$.
Statement 3: The planes $P$ and $P ^ { \prime }$ are secant and their intersection is the line $\Delta$
$$\text { with a parametric representation: } \left\{ \begin{array} { l } x = 3 - 3 t \\ y = 0 \\ z = t \end{array} , t \in \mathbf { R } \right. \text {. }$$
Statement 4: The point $J$ is the orthogonal projection of the point $H$ onto the line ( $C D$ ).

Let $L$ be the line of intersection of the planes $x + y = 0$ and $y + z = 0$.
(a) Write the vector equation of $L$, i.e., find $(a, b, c)$ and $(p, q, r)$ such that $$L = \{(a, b, c) + \lambda(p, q, r) \mid \lambda \text{ is a real number.}\}$$ (b) Find the equation of a plane obtained by rotating $x + y = 0$ about $L$ by $45^\circ$.

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 $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024\}$. [1 point]
Options:

1. The empty set
2. A singleton set
3. A line
4. A pair of lines
5. A circle
6. A plane perpendicular to $\mathbf{u}$
7. A plane parallel to $\mathbf{u}$
8. An infinite cone
9. A finite cone
10. A sphere
11. None of the above

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points]
Options:

1. The empty set
2. A singleton set
3. A line
4. A pair of lines
5. A circle
6. A plane perpendicular to $\mathbf{u}$
7. A plane parallel to $\mathbf{u}$
8. An infinite cone
9. A finite cone
10. A sphere
11. None of the above

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024(\mathbf{v} \cdot \mathbf{v})\}$. [2 points]
Options:

1. The empty set
2. A singleton set
3. A line
4. A pair of lines
5. A circle
6. A plane perpendicular to $\mathbf{u}$
7. A plane parallel to $\mathbf{u}$
8. An infinite cone
9. A finite cone
10. A sphere
11. None of the above

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) $\sqrt { 11 }$

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]

In coordinate space, the equation of the plane that is perpendicular to the line $\frac { x - 2 } { 2 } = \frac { y - 2 } { 3 } = z - 1$ and passes through the point $( 1 , - 5,2 )$ is $2 x + a y + b z + c = 0$. Find the value of $a + b + c$. [3 points]

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, 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, let $\theta$ be the acute angle between the plane $2 x + 2 y - z + 5 = 0$ and the $xy$-plane. What is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

The plane $E : 3 x _ { 1 } + 2 x _ { 2 } + 2 x _ { 3 } = 6$ contains a point whose three coordinates are equal. Determine these coordinates.

The point $L$, which lies vertically above the midpoint of the edge $\left[ \mathrm { A } _ { 1 } \mathrm {~A} _ { 2 } \right]$, represents the position of a floodlight in the model, which is installed 12 m above the base. The points $L , B _ { 2 }$ and $B _ { 3 }$ determine a plane $F$. Find an equation of $F$ in normal form.
(for verification: $F : 3 x _ { 1 } + x _ { 2 } + 5 x _ { 3 } - 90 = 0$ )

The plane $F$ intersects the $x _ { 1 } x _ { 2 }$-plane in the line $g$. Determine an equation of $g$.
(for verification: $g : \vec { X } = \left( \begin{array} { c } 30 \\ 0 \\ 0 \end{array} \right) + \lambda \cdot \left( \begin{array} { c } 1 \\ - 3 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$ )

A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality.
Determine the diameter of the marble sphere to the nearest centimeter.

Determine an equation of $E$ in coordinate form. (for verification: $E : x _ { 1 } + x _ { 2 } + 2 x _ { 3 } - 20 = 0$ )

Justify that every line of the family lies in $E$, and determine the value $k$ for which the point $C$ lies on $g _ { k }$. (for verification: $k = 0.8$ )

Justify that the magnitude of the angle of intersection of $g _ { k }$ and the $x _ { 1 } x _ { 2 }$-plane is less than $30 ^ { \circ }$ if $2 k ^ { 2 } > 1$ holds.

Let $v_1$ and $v_2$ be two vectors of $\mathcal{H} = \{v\in V \mid B(v,v)=-1 \text{ and } z_v > 0\}$. Show that $$B(v_1,v_2) \leq -1,$$ with equality if and only if $v_1 = v_2$.