This exercise is a multiple choice questionnaire. For each question, three answers are proposed and only one of them is correct. The candidate will write on the answer sheet the number of the question followed by the chosen answer and will justify their choice. One point is awarded for each correct and properly justified answer. An unjustified answer will not be taken into account. No points are deducted in the absence of an answer or in case of an incorrect answer.
For questions 1 and 2, space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The line $\mathscr{D}$ is defined by the parametric representation $\left\{\begin{array}{rl} x &= 5-2t \\ y &= 1+3t \\ z &= 4 \end{array},\, t \in \mathbb{R}\right.$.

1. We denote by $\mathscr{P}$ the plane with Cartesian equation $3x + 2y + z - 6 = 0$. a. The line $\mathscr{D}$ is perpendicular to the plane $\mathscr{P}$. b. The line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$. c. The line $\mathscr{D}$ is contained in the plane $\mathscr{P}$.
2. We denote by $\mathscr{D}'$ the line that passes through point A with coordinates $(3;1;1)$ and has direction vector $\vec{u} = 2\vec{i} - \vec{j} + 2\vec{k}$. a. The lines $\mathscr{D}$ and $\mathscr{D}'$ are parallel. b. The lines $\mathscr{D}$ and $\mathscr{D}'$ are secant. c. The lines $\mathscr{D}$ and $\mathscr{D}'$ are not coplanar.

For questions 3 and 4, the plane is equipped with a direct orthonormal coordinate system with origin O.

1. Let $\mathscr{E}$ be the set of points $M$ with affix $z$ satisfying $|z + \mathrm{i}| = |z - \mathrm{i}|$. a. $\mathscr{E}$ is the $x$-axis. b. $\mathscr{E}$ is the $y$-axis. c. $\mathscr{E}$ is the circle with center O and radius 1.
2. We denote by B and C two points in the plane whose respective affixes $b$ and $c$ satisfy the equality $\dfrac{c}{b} = \sqrt{2}\,\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$. a. The triangle OBC is isosceles with apex O. b. The points O, B, C are collinear. c. The triangle OBC is isosceles and right-angled at B.

This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer is worth 1 point. An incorrect answer or no answer does not deduct any points.

1. Let $z _ { 1 } = \sqrt { 6 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$ and $z _ { 2 } = \sqrt { 2 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. The exponential form of $\mathrm { i } \frac { z _ { 1 } } { z _ { 2 } }$ is: a. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 19 \pi } { 12 } }$ b. $\sqrt { 12 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 12 } }$ c. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 7 \pi } { 12 } }$ d. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 13 \pi } { 12 } }$
2. The equation $- z = \bar { z }$, with unknown complex number $z$, admits: a. one solution b. two solutions c. infinitely many solutions whose image points in the complex plane are located on a line. d. infinitely many solutions whose image points in the complex plane are located on a circle.
3. In a coordinate system of space, consider the three points $A ( 1 ; 2 ; 3 ) , B ( - 1 ; 5 ; 4 )$ and $C ( - 1 ; 0 ; 4 )$. The line parallel to the line $( A B )$ passing through point $C$ has the parametric representation: a. $\left\{ \begin{array} { l } x = - 2 t - 1 \\ y = 3 t \\ z = t + 4 \end{array} , t \in \mathbb { R } \right.$ b. $\left\{ \begin{array} { l } x = - 1 \\ y = 7 t \\ z = 7 t + 4 \end{array} , t \in \mathbb { R } \right.$ c. $\left\{ \begin{array} { l } x = - 1 - 2 t \\ y = 5 + 3 t \\ z = 4 + t \end{array} , t \in \mathbb { R } \right.$ d. $\left\{ \begin{array} { l } x = 2 t \\ y = - 3 t \\ z = - t \end{array} , t \in \mathbb { R } \right.$
4. In an orthonormal coordinate system of space, consider the plane $\mathscr { P }$ passing through point $D ( - 1 ; 2 ; 3 )$ and with normal vector $\vec { n } ( 3 ; - 5 ; 1 )$, and the line $\Delta$ with parametric representation $\left\{ \begin{array} { l } x = t - 7 \\ y = t + 3 \\ z = 2 t + 5 \end{array} , t \in \mathbb { R } \right.$. a. The line $\Delta$ is perpendicular to the plane $\mathscr { P }$. b. The line $\Delta$ is parallel to the plane $\mathscr { P }$ and has no common point with the plane $\mathscr { P }$. c. The line $\Delta$ and the plane $\mathscr { P }$ are secant. d. The line $\Delta$ is contained in the plane $\mathscr { P }$.

Exercise 3 -- Common to all candidates
The four questions in this exercise are independent. For each question, a statement is proposed. Indicate whether each of them is true or false, by justifying your answer. An unjustified answer earns no points. In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(O, \vec{u}, \vec{v})$. Consider the points A, B, C, D and E with complex numbers respectively: $$a = 2 + 2\mathrm{i}, \quad b = -\sqrt{3} + \mathrm{i}, \quad c = 1 + \mathrm{i}\sqrt{3}, \quad d = -1 + \frac{\sqrt{3}}{2}\mathrm{i} \quad \text{and} \quad e = -1 + (2 + \sqrt{3})\mathrm{i}.$$

1. Statement 1: the points A, B and C are collinear.
2. Statement 2: the points B, C and D belong to the same circle with center E.
3. In this question, space is equipped with a coordinate system $(O, \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider the points $I(1; 0; 0)$, $J(0; 1; 0)$ and $K(0; 0; 1)$. Statement 3: the line $\mathscr{D}$ with parametric representation $\left\{\begin{aligned} x &= 2 - t \\ y &= 6 - 2t \\ z &= -2 + t \end{aligned}\right.$ where $t \in \mathbb{R}$, intersects the plane (IJK) at point $E\left(-\frac{1}{2}; 1; \frac{1}{2}\right)$.
4. In the cube ABCDEFGH, the point T is the midpoint of segment $[HF]$. Statement 4: the lines (AT) and (EC) are orthogonal.

For each of the four propositions below, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. No answer is not penalized.

1. Proposition 1: In the plane with an orthonormal coordinate system, the set of points $M$ whose affix $z$ satisfies the equality $| z - \mathrm { i } | = | z + 1 |$ is a line.
2. Proposition 2: The complex number $( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 4 }$ is a real number.
3. Let ABCDEFGH be a cube. Proposition 3: The lines (EC) and (BG) are orthogonal.
4. Space is equipped with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$). Let the plane $\mathscr { P }$ with Cartesian equation $x + y + 3z + 4 = 0$. We denote S the point with coordinates $( 1 , - 2 , - 2 )$. Proposition 4: The line passing through S and perpendicular to the plane $\mathscr { P }$ has parametric representation $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.

The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$. We denote by i the complex number such that $\mathrm { i } ^ { 2 } = - 1$. We consider the point A with affixe $z _ { \mathrm { A } } = 1$ and the point B with affixe $z _ { \mathrm { B } } = \mathrm { i }$. To any point $M$ with affixe $z _ { M } = x + \mathrm { i } y$, with $x$ and $y$ two real numbers such that $y \neq 0$, we associate the point $M ^ { \prime }$ with affixe $z _ { M ^ { \prime } } = - \mathrm { i } z _ { M }$. We denote by $I$ the midpoint of the segment [AM]. The purpose of the exercise is to show that for any point $M$ not belonging to (OA), the median $( \mathrm { O } I )$ of the triangle $\mathrm { OA} M$ is also a height of the triangle $\mathrm { OB } M ^ { \prime }$ (property 1) and that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$ (property 2).

1. In this question and only in this question, we take $z _ { M } = 2 \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. a. Determine the algebraic form of $z _ { M }$. b. Show that $z _ { M ^ { \prime } } = - \sqrt { 3 } - \mathrm { i }$. Determine the modulus and an argument of $z _ { M ^ { \prime } }$. c. Place the points $\mathrm { A } , \mathrm { B } , M , M ^ { \prime }$ and $I$ in the coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$ using 2 cm as the graphical unit. Draw the line $( \mathrm { O } I )$ and quickly verify properties 1 and 2 using the graph.
2. We return to the general case by taking $z _ { M } = x + \mathrm { i } y$ with $y \neq 0$. a. Determine the affixe of point $I$ as a function of $x$ and $y$. b. Determine the affixe of point $M ^ { \prime }$ as a function of $x$ and $y$. c. Write the coordinates of points $I$, B and $M ^ { \prime }$. d. Show that the line $( \mathrm { O } I )$ is a height of the triangle $\mathrm { OB } M ^ { \prime }$. e. Show that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$.

Exercise 4 — For candidates who have NOT followed the specialization course
The plane is referred to an orthonormal direct coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$.
Let $\mathbb { C }$ denote the set of complex numbers.
For each of the following propositions, state whether it is true or false by justifying the answer.
1. Proposition: For every natural number $n$ : $( 1 + \mathrm { i } ) ^ { 4 n } = ( - 4 ) ^ { n }$.
2. Let (E) be the equation $( z - 4 ) \left( z ^ { 2 } - 4 z + 8 \right) = 0$ where $z$ denotes a complex number.
Proposition: The points whose affixes are the solutions, in $\mathbb { C }$, of (E) are the vertices of a triangle with area 8.
3. Proposition: For every real number $\alpha , 1 + \mathrm { e } ^ { 2 i \alpha } = 2 \mathrm { e } ^ { \mathrm { i } \alpha } \cos ( \alpha )$.
4. Let A be the point with affix $z _ { \mathrm { A } } = \frac { 1 } { 2 } ( 1 + \mathrm { i } )$ and $M _ { n }$ the point with affix $\left( z _ { \mathrm { A } } \right) ^ { n }$ where $n$ denotes a natural number greater than or equal to 2.
Proposition: if $n - 1$ is divisible by 4, then the points O, A and $M _ { n }$ are collinear.
5. Let j be the complex number with modulus 1 and argument $\frac { 2 \pi } { 3 }$.
Proposition: $1 + \mathrm { j } + \mathrm { j } ^ { 2 } = 0$.

For each of the four following statements, indicate whether it is true or false by justifying your answer.
One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.

1. In the plane with an orthonormal coordinate system, let $S$ denote the set of points $M$ whose affix $z$ satisfies the two conditions: $$|z - 1| = |z - \mathrm{i}| \quad \text{and} \quad |z - 3 - 2\mathrm{i}| \leqslant 2.$$ In the figure below, we have represented the circle with center at the point with coordinates $(3;2)$ and radius 2, and the line with equation $y = x$. This line intersects the circle at two points A and B.
   Statement 1: the set $S$ is the segment $[AB]$.
   
2. Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
   
3. For questions 3 and 4, consider the points $\mathrm{E}(2; 1; -3)$, $\mathrm{F}(1; -1; 2)$ and $\mathrm{G}(-1; 3; 1)$ whose coordinates are defined in an orthonormal coordinate system of space.
   Statement 3: a parametric representation of the line $(EF)$ is given by: $$\left\{\begin{array}{rlr} x & = & 2t \\ y & = & -3 + 4t, \quad t \in \mathbb{R} \\ z & = 7 - 10t \end{array}\right.$$
   
4. Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.

We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For each of the three following propositions, indicate whether it is true or false and justify the chosen answer. One point is awarded for each correct answer properly justified. An unjustified answer is not taken into account.
Proposition 1 The set of points in the plane with affixe $z$ such that $| z - 4 | = | z + 2 \mathrm { i } |$ is a line that passes through the point A with affixe 3i.
Proposition 2 Let ( $E$ ) be the equation $( z - 1 ) \left( z ^ { 2 } - 8 z + 25 \right) = 0$ where $z$ belongs to the set $\mathbb { C }$ of complex numbers. The points in the plane whose affixes are the solutions in $\mathbb { C }$ of the equation ( $E$ ) are the vertices of a right triangle.
Proposition 3 $\frac { \pi } { 3 }$ is an argument of the complex number $( - \sqrt { 3 } + \mathrm { i } ) ^ { 8 }$.

Exercise 2 (4 points)

The complex plane is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. To every point $M$ with affixe $z$, we associate the point $M'$ with affixe $$z' = -z^2 + 2z$$ The point $M'$ is called the image of point $M$.

1. Solve in the set $\mathbb{C}$ of complex numbers the equation: $$-z^2 + 2z - 2 = 0$$ Deduce the affixes of the points whose image is the point with affixe 2.
   
2. Let $M$ be a point with affixe $z$ and $M'$ its image with affixe $z'$.
   We denote $N$ the point with affixe $z_N = z^2$. Show that $M$ is the midpoint of segment $[NM']$.
   
3. In this question, we assume that the point $M$ with affixe $z$ belongs to the circle $\mathscr{C}$ with center O and radius 1. We denote $\theta$ an argument of $z$. a. Determine the modulus of each of the complex numbers $z$ and $z_N$, as well as an argument of $z_N$ as a function of $\theta$. b. On the figure given in the appendix on page 7, a point $M$ on the circle $\mathscr{C}$ has been represented. Construct on this figure the points $N$ and $M'$ using a ruler and compass (leave the construction lines visible). c. Let A be the point with affixe 1. What is the nature of triangle $AMM'$?

Exercise 3 -- Common to all candidates

In a vast plain, a network of sensors makes it possible to detect lightning and to produce an image of storm phenomena. The radar screen is divided into forty sectors denoted by a letter and a number between 1 and 5. The lightning sensor is represented by the center of the screen; five concentric circles corresponding to the respective radii $20, 40, 60, 80$ and 100 kilometres delimit five zones numbered from 1 to 5, and eight segments starting from the sensor delimit eight portions of the same angular opening, named in the trigonometric direction from A to H.
We assimilate the radar screen to a part of the complex plane by defining an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$:

• the origin O marks the position of the sensor;
• the abscissa axis is oriented from West to East;
• the ordinate axis is oriented from South to North;
• the chosen unit is the kilometre.

In the following, a point on the radar screen is associated with a point with affix $z$.

Part A

1. We denote $z_{P}$ the affix of the point P located in sector B3 on the graph. We call $r$ the modulus of $z_{P}$ and $\theta$ its argument in the interval $]-\pi ; \pi]$. Among the four following propositions, determine the only one that proposes a correct bound for $r$ and for $\theta$ (no justification is required):
   

   Proposition A	Proposition B	Proposition C	Proposition D
   \begin{tabular}{ c } $40 < r < 60$
   and
   $0 < \theta < \frac{\pi}{4}$

   &

   $20 < r < 40$

   and

   $\frac{\pi}{2} < \theta < \frac{3\pi}{4}$

   &

   $40 < r < 60$

   and

   $\frac{\pi}{4} < \theta < \frac{\pi}{2}$

   &

   $0 < r < 60$

   and

   $-\frac{\pi}{2} < \theta < -\frac{\pi}{4}$

   \hline \end{tabular}
   
2. A lightning impact is materialized on the screen at a point with affix $z$. In each of the two following cases, determine the sector to which this point belongs: a. $z = 70 \mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}$; b. $z = -45\sqrt{3} + 45\mathrm{i}$.

Part B

We assume in this part that the sensor displays an impact at point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. When the sensor displays the impact point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, the affix $z$ of the actual lightning impact point admits:

• a modulus that can be modeled by a random variable $M$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 5$;
• an argument that can be modeled by a random variable $T$ following a normal distribution with mean $\frac{\pi}{3}$ and standard deviation $\frac{\pi}{12}$.

We assume that the random variables $M$ and $T$ are independent. In the following, probabilities will be rounded to $10^{-3}$ near.

1. Calculate the probability $P(M < 0)$ and interpret the result obtained.
2. Calculate the probability $P(M \in ]40 ; 60[)$.
3. We admit that $P\left(T \in \left]\frac{\pi}{4} ; \frac{\pi}{2}\right[\right) = 0.819$. Deduce the probability that the lightning actually struck sector B3 according to this modeling.

Exercise 4 (3 points)
In the complex plane equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$, we consider the points A and B with complex numbers respectively $z_{\mathrm{A}} = 2\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_{\mathrm{B}} = 2\mathrm{e}^{\mathrm{i}\frac{3\pi}{4}}$.

1. Show that OAB is a right isosceles triangle.
2. We consider the equation $$(E): z^2 - \sqrt{6}\, z + 2 = 0$$ Show that one of the solutions of $(E)$ is the complex number of a point located on the circumscribed circle of triangle OAB.

Consider in $\mathbb{C}$ the equation: $$\left(4z^2 - 20z + 37\right)(2z - 7 + 2i) = 0$$ Statement 4: the solutions of the equation are the affixes of points belonging to the same circle with centre the point P with affix 2. Indicate whether Statement 4 is true or false, justifying your answer.

Specify whether each of the following statements is true or false by justifying your answer.

1. Let $m$ be a real number and let the equation $( E )$ : $2 z ^ { 2 } + ( m - 5 ) z + m = 0$. a. Statement 1 : ``For $m = 4$, the equation ( $E$ ) admits two real solutions.'' b. Statement 2 : ``There exists only one value of $m$ such that ( $E$ ) admits two complex solutions that are pure imaginary numbers.''
2. In the complex plane, we consider the set $S$ of points $M$ with affixe $z$ satisfying: $$| z - 6 | = | z + 5 i |$$ Statement 3 : ``The set $S$ is a circle.''
3. We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We denote $d$ the line with parametric representation: $$d : \left\{ \begin{aligned} x & = - 1 + t \\ y & = 2 - t \quad t \in \mathbb { R } . \\ z & = 3 + t \end{aligned} \right.$$ We denote $d ^ { \prime }$ the line passing through the point $\mathrm { B } ( 4 ; 4 ; - 6 )$ and with direction vector $\vec { v } ( 5 ; 2 ; - 9 )$. Statement 4 : ``The lines $d$ and $d ^ { \prime }$ are coplanar.''
4. We consider the cube ABCDEFGH. Statement 5 : ``The vector $\overrightarrow { \mathrm { DE } }$ is a normal vector to the plane (ABG).''

Let $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { 13 }$ be real numbers and let $A$ be the average of the complex numbers $e ^ { i \theta _ { 1 } } , e ^ { i \theta _ { 2 } } \ldots , e ^ { i \theta _ { 13 } }$, where $i = \sqrt { - 1 }$. As the values of $\theta$'s vary over all 13-tuples of real numbers, find (i) the maximum value attained by $| A |$, (ii) the minimum value attained by $| A |$.

Answer the three questions below. To answer (i) and (ii), replace ? with exactly one of the following four options: $<$, $=$, $>$, not enough information to compare.
(i) Suppose $z_1, z_2$ are complex numbers. One of them is in the second quadrant and the other is in the third quadrant. Then $\left|\left|z_1\right| - \left|z_2\right|\right| \quad ? \quad \left|z_1 + z_2\right|$.
(ii) Complex numbers $z_1$, $z_2$ and $0$ form an equilateral triangle. Then $\left|z_1^2 + z_2^2\right| \quad ? \quad \left|z_1 z_2\right|$.
(iii) Let $1, z_1, z_2, z_3, z_4, z_5, z_6, z_7$ be the complex 8th roots of unity. Find the value of $\prod_{i=1,\ldots,7}(1 - z_i)$, where the symbol $\Pi$ denotes product.

You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$.
Statements
(5) There is exactly one such triangle T that is equilateral. (6) There are exactly two such triangles $T$ that are right angled isosceles. (7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles. (8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.

This question is about complex numbers.
Statements
(9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant. (10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0. (11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions. (12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.

Let complex numbers $z _ { 1 } , z _ { 2 }$ satisfy $\left| z _ { 1 } \right| = \left| z _ { 2 } \right| = 2 , z _ { 1 } + z _ { 2 } = \sqrt { 3 } + \mathrm { i }$ , then $\left| z _ { 1 } - z _ { 2 } \right| = $ $\_\_\_\_$.

Represent on the same figure $\tau _ { 0 } , \tau _ { 1 } , \tau$.

Throughout the problem, the set $\mathbf { C }$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2 = -1$). We denote by $K$ the set of triplets $(\alpha, \beta, \gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha + \beta + \gamma = 1$. If $(a,b,c) \in \mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma) \in K\}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
a) Let $a \in \mathbf{C}$ and $\theta \in \mathbf{R}$. Prove that the image $z'$ of the complex number $z$ by the reflection whose axis is the line passing through $a$ and directed by $\mathrm{e}^{\mathrm{i}\theta}$ satisfies the relation: $$z' - a = \mathrm{e}^{2\mathrm{i}\theta} \overline{(z-a)}$$
b) Establish a relation analogous to that of the previous question between a complex number $z$ and its image $z'$ by the homothety with center $a$ and ratio $\rho > 0$.
c) Prove that $\phi_0$ is the composition of a reflection whose axis we will specify and a homothety with strictly positive ratio to be specified and whose center belongs to the axis of the reflection. Prove an analogous property for $\phi_1$. Are these decompositions unique?

Throughout the problem, the set $\mathbf{C}$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2=-1$). We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
What is the image of a filled triangle $\widehat{abc}$ by $\phi_0$ and by $\phi_1$? Determine $\phi_0(\tau)$ and $\phi_1(\tau)$.

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
Let $(r_n)_{n\geq 1}$ be an element of $\{0,1\}^{\mathbf{N}^*}$. For each non-zero natural number $n$, we denote $\tilde{\tau}_n = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_n}(\tau)$.
Show that $\bigcap_{n\geq 1}\tilde{\tau}_n$ is reduced to a single point belonging to $\tau$.

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$.
Determine the unique element $f_0$ of $\mathcal{E}$ which is affine.

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. If $g\in\mathcal{E}$, we denote by $Tg$ the map from $[0,1]$ to $\mathbf{C}$ defined by: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$
Show that $Tg\in\mathcal{E}$ for all $g\in\mathcal{E}$.

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$. The norm of uniform convergence on the $\mathbf{C}$-vector space of continuous maps from $[0,1]$ to $\mathbf{C}$ is denoted $\|\cdot\|_\infty$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. If $g\in\mathcal{E}$, we denote by $Tg$ the map from $[0,1]$ to $\mathbf{C}$ defined by: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$
Let $g_1$ and $g_2$ be two elements of $\mathcal{E}$. Prove that: $$\|Tg_2 - Tg_1\|_\infty = \frac{1}{\sqrt{2}}\|g_2 - g_1\|_\infty$$