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 the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer earns no points. An absence of an answer is not penalized.
In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. We denote by $\mathbb{R}$ the set of real numbers.
Statement 1: The point with affix $(-1+i)^{10}$ is located on the imaginary axis.
Statement 2: In the set of complex numbers, the equation $$z - \bar{z} + 2 - 4\mathrm{i} = 0$$ admits a unique solution.
Statement 3: $$\ln\left(\sqrt{\mathrm{e}^{7}}\right) + \frac{\ln\left(\mathrm{e}^{9}\right)}{\ln\left(\mathrm{e}^{2}\right)} = \frac{\mathrm{e}^{\ln 2 + \ln 3}}{\mathrm{e}^{\ln 3 - \ln 4}}$$
Statement 4: $$\int_{0}^{\ln 3} \frac{\mathrm{e}^{x}}{\mathrm{e}^{x}+2}\,\mathrm{d}x = -\ln\left(\frac{3}{5}\right)$$
Statement 5: The equation $\ln(x-1) - \ln(x+2) = \ln 4$ admits a unique solution in $\mathbb{R}^{*}$.

1. Solve in the set $\mathbb { C }$ of complex numbers the equation (E) with unknown $z$ : $$z ^ { 2 } - 8 z + 64 = 0$$
The complex plane is equipped with a direct orthonormal reference frame $( \mathrm { O } ; \vec { u } , \vec { v } )$.
2. We consider the points $\mathrm { A } , \mathrm { B }$ and C with affixes respectively $a = 4 + 4 \mathrm { i } \sqrt { 3 }$, $b = 4 - 4 \mathrm { i } \sqrt { 3 }$ and $c = 8 \mathrm { i }$. a. Calculate the modulus and an argument of the number $a$. b. Give the exponential form of the numbers $a$ and $b$. c. Show that the points $\mathrm { A } , \mathrm { B }$ and C lie on the same circle with center O whose radius will be determined. d. Place the points $\mathrm { A } , \mathrm { B }$ and C in the reference frame ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For the rest of the exercise, you may use the figure from question 2. d. completed as the questions progress.
3. We consider the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ with affixes respectively $a ^ { \prime } = a \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } } , b ^ { \prime } = b \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and $c ^ { \prime } = c \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$. a. Show that $b ^ { \prime } = 8$. b. Calculate the modulus and an argument of the number $a ^ { \prime }$.
For the rest we admit that $a ^ { \prime } = - 4 + 4 \mathrm { i } \sqrt { 3 }$ and $c ^ { \prime } = - 4 \sqrt { 3 } + 4 \mathrm { i }$.
4. We admit that if $M$ and $N$ are two points in the plane with affixes respectively $m$ and $n$ then the midpoint $I$ of the segment $[ M N ]$ has affix $\frac { m + n } { 2 }$ and the length $M N$ is equal to $| n - m |$. a. We denote $r , s$ and $t$ the affixes of the midpoints respectively $\mathrm { R } , \mathrm { S }$ and T of the segments $\left[ \mathrm { A } ^ { \prime } \mathrm { B } \right] , \left[ \mathrm { B } ^ { \prime } \mathrm { C } \right]$ and $\left[ \mathrm { C } ^ { \prime } \mathrm { A } \right]$. Calculate $r$ and $s$. We admit that $t = 2 - 2 \sqrt { 3 } + \mathrm { i } ( 2 + 2 \sqrt { 3 } )$. b. What conjecture can be made about the nature of triangle RST? Justify this result.

Exercise 4 — Candidates who have not chosen the specialty course

The plane is equipped with the direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$. We are given the complex number $\mathrm{j} = -\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$. The purpose of this exercise is to study some properties of the number j and to highlight a connection between this number and equilateral triangles.

Part A: properties of the number j

1. a. Solve in the set $\mathbb{C}$ of complex numbers the equation $$z^{2} + z + 1 = 0$$ b. Verify that the complex number j is a solution of this equation.
2. Determine the modulus and an argument of the complex number j, then give its exponential form.
3. Prove the following equalities: a. $j^{3} = 1$; b. $\mathrm{j}^{2} = -1 - \mathrm{j}$.
4. Let $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ be the images respectively of the complex numbers $1, \mathrm{j}$ and $\mathrm{j}^{2}$ in the plane.
   What is the nature of triangle PQR? Justify the answer.

Part B

Let $a, b, c$ be three complex numbers satisfying the equality $a + \mathrm{j}b + \mathrm{j}^{2}c = 0$. Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ denote the images respectively of the numbers $a, b, c$ in the plane.

1. Using question A-3.b., prove the equality: $a - c = \mathrm{j}(c - b)$.
2. Deduce that $\mathrm{AC} = \mathrm{BC}$.
3. Prove the equality: $a - b = \mathrm{j}^{2}(b - c)$.
4. Deduce that triangle ABC is equilateral.

We want to model in the plane the shell of a nautilus using a broken line in the form of a spiral. We are interested in the area delimited by this line.
We equip the plane with a direct orthonormal coordinate system $(O; \vec{u}; \vec{v})$. Let $n$ be an integer greater than or equal to 2. For all integer $k$ ranging from 0 to $n$, we define the complex numbers $z_k = \left(1 + \dfrac{k}{n}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{n}}$ and we denote by $M_k$ the point with affix $z_k$. In this model, the perimeter of the nautilus is the broken line connecting all the points $M_k$ with $0 \leqslant k \leqslant n$.
Part A: Broken line formed from seven points
In this part, we assume that $n = 6$. Thus, for $0 \leqslant k \leqslant 6$, we have $z_k = \left(1 + \dfrac{k}{6}\right) \mathrm{e}^{\mathrm{i}\frac{2k\pi}{6}}$.

1. Determine the algebraic form of $z_1$.
2. Verify that $z_0$ and $z_6$ are integers that you will determine.
3. Calculate the length of the altitude from $M_1$ in the triangle $OM_0M_1$ then establish that the area of this triangle is equal to $\dfrac{7\sqrt{3}}{24}$.

Part B: Broken line formed from $n+1$ points
In this part, $n$ is an integer greater than or equal to 2.

1. For all integer $k$ such that $0 \leqslant k \leqslant n$, determine the length $OM_k$.
2. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, determine a measure of the angles $(\vec{u}; \overrightarrow{OM_k})$ and $(\vec{u}; \overrightarrow{OM_{k+1}})$. Deduce a measure of the angle $(\overrightarrow{OM_k}; \overrightarrow{OM_{k+1}})$.
3. For $k$ an integer such that $0 \leqslant k \leqslant n-1$, calculate the area of the triangle $OM_kM_{k+1}$ as a function of $n$ and $k$.

Exercise 2

4 points
We consider the complex numbers $z_{n}$ defined for every integer $n \geqslant 0$ by the value of $z_{0}$, where $z_{0}$ is different from 0 and 1, and the recurrence relation:
$$z_{n+1} = 1 - \frac{1}{z_{n}}$$

1. a. In this question, we assume that $z_{0} = 2$. Determine the numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$ b. In this question, we assume that $z_{0} = \mathrm{i}$. Determine the algebraic form of the complex numbers $z_{1}, z_{2}, z_{3}, z_{4}, z_{5}$ and $z_{6}$. c. In this question we return to the general case where $z_{0}$ is a given complex number. What can we conjecture about the values taken by $z_{3n}$ according to the values of the natural integer $n$? Prove this conjecture.
2. Determine $z_{2016}$ in the case where $z_{0} = 1 + \mathrm{i}$.
3. Are there values of $z_{0}$ such that $z_{0} = z_{1}$? What can we say about the sequence $(z_{n})$ in this case?

Exercise 4 - Candidates who have NOT followed the specialization course

For each of the five following propositions, indicate whether it is true or false and justify the answer chosen. One point is awarded for each correct answer correctly justified. An unjustified answer is not taken into account. An absence of answer is not penalized.

Proposition 1:

In the complex plane equipped with an orthonormal coordinate system, the points A, B and C with affixes respectively $z _ { \mathrm { A } } = \sqrt { 2 } + 3 \mathrm { i } , z _ { \mathrm { B } } = 1 + \mathrm { i }$ and $z _ { \mathrm { C } } = - 4 \mathrm { i }$ are not collinear.

Proposition 2:

There does not exist a non-zero natural integer $n$ such that $[ \mathrm { i } ( 1 + \mathrm { i } ) ] ^ { 2n }$ is a strictly positive real number.

Proposition 3:

ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The section of the cube by the plane (BDL) is a triangle.

Proposition 4:

ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The triangle DBL is right-angled at B.

Proposition 5:

We consider the function $f$ defined on the interval [2;5] and whose variation table is given below:

$x$	2	3	4	5
\begin{tabular}{ c } Variations
$\operatorname { of } f$

& 3 & & & 2 & & & 0 & 1 \end{tabular}
The integral $\int _ { 2 } ^ { 5 } f ( x ) \mathrm { d } x$ is between 1,5 and 6.

Exercise 4

Common to all candidates

The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D distinct with complex numbers $z _ { \mathrm { A } } , z _ { \mathrm { B } } , z _ { \mathrm { C } }$ and $z _ { \mathrm { D } }$ such that:
$$\left\{ \begin{array} { l } z _ { \mathrm { A } } + z _ { \mathrm { C } } = z _ { \mathrm { B } } + z _ { \mathrm { D } } \\ z _ { \mathrm { A } } + \mathrm { i } z _ { \mathrm { B } } = z _ { \mathrm { C } } + \mathrm { i } z _ { \mathrm { D } } \end{array} \right.$$
Prove that the quadrilateral ABCD is a square.

In this exercise, $x$ and $y$ are real numbers greater than 1. In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ), we consider the points $\mathrm { A} , \mathrm { B }$ and C with affixes respectively $z_{\mathrm{A}} = 1 + \mathrm{i}$, $z_{\mathrm{B}} = x + \mathrm{i}$, $z_{\mathrm{C}} = y + \mathrm{i}$.
Problem: we seek the possible values of real numbers $x$ and $y$, greater than 1, for which :
$$\mathrm { OC } = \mathrm { OA } \times \mathrm { OB } \quad \text { and } ( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$$

1. Prove that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$.
2. Reproduce on your answer sheet and complete the following algorithm so that it displays all couples $( x , y )$ such that : \begin{verbatim} For x going from 1 to ... do For... If... Display x and y End If End For End For \end{verbatim} When this algorithm is executed, it displays the value 2 for variable $x$ and the value 3 for variable $y$.
3. Study of a particular case: in this question only, we take $x = 2$ and $y = 3$. a. Give the modulus and an argument of $z _ { \mathrm { A } }$. b. Show that $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$. c. Show that $z _ { \mathrm { B } } z _ { \mathrm { C } } = 5 z _ { \mathrm { A } }$ and deduce that $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$.
4. We return to the general case, and we seek whether there exist other values of real numbers $x$ and $y$ such that points $\mathrm { A } , \mathrm { B }$ and C satisfy the two conditions : $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$ and $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$. Recall that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$ (question 1.). a. Prove that if $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$, then $\arg \left[ \frac { ( x + \mathrm { i } ) ( y + \mathrm { i } ) } { 1 + \mathrm { i } } \right] = 0 \bmod 2 \pi$.
   Deduce that under this condition : $x + y - x y + 1 = 0$. b. Prove that if the two conditions are satisfied and moreover $x \neq 1$, then :
   $$y = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } y = \frac { x + 1 } { x - 1 }$$
   
5. We define the functions $f$ and $g$ on the interval $] 1 ; + \infty [$ by :
   $$f ( x ) = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } g ( x ) = \frac { x + 1 } { x - 1 }$$
   Determine the number of solutions to the initial problem. We may use the function $h$ defined on the interval $] 1 ; + \infty [$ by $h ( x ) = f ( x ) - g ( x )$ and rely on the screenshot of a computer algebra software given below.

Exercise 4 — Candidates who have not followed the specialization course
We define the sequence of complex numbers $( z _ { n } )$ in the following way: $z _ { 0 } = 1$ and, for every natural integer $n$,
$$z _ { n + 1 } = \frac { 1 } { 3 } z _ { n } + \frac { 2 } { 3 } \mathrm { i } .$$
We place ourselves in a plane with an orthonormal direct coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. For every natural integer $n$, we denote $\mathrm { A } _ { n }$ the point in the plane with affix $z _ { n }$. For every natural integer $n$, we set $u _ { n } = z _ { n } - \mathrm { i }$ and we denote $\mathrm { B } _ { n }$ the point with affix $u _ { n }$. We denote C the point with affix i.

1. Express $u _ { n + 1 }$ as a function of $u _ { n }$, for every natural integer $n$.
2. Prove that, for every natural integer $n$,

$$u _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n } ( 1 - \mathrm { i } ) .$$

\setcounter{enumi}{2}
1. a. For every natural integer $n$, calculate, as a function of $n$, the modulus of $u _ { n }$. b. Prove that

$$\lim _ { n \rightarrow + \infty } \left| z _ { n } - \mathrm { i } \right| = 0$$
c. What geometric interpretation can be given of this result?
4. a. Let $n$ be a natural integer. Determine an argument of $u _ { n }$. b. Prove that, as $n$ ranges over the set of natural integers, the points $\mathrm { B } _ { n }$ are collinear. c. Prove that, for every natural integer $n$, the point $\mathrm { A } _ { n }$ belongs to the line with reduced equation:
$$y = - x + 1 .$$

The plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$.
The purpose of this exercise is to determine the non-zero complex numbers $z$ such that the points with affixes $1$, $z^2$ and $\dfrac{1}{z}$ are collinear. On the graph provided in the appendix, point A has affix 1.
Part A: study of examples
1. A first example
In this question, we set $z = \mathrm{i}$. a. Give the algebraic form of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_1$ with affix $z^2$, and $P_1$ with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_1$ and $P_1$ are not collinear.
2. An equation
Solve in the set of complex numbers the equation with unknown $z$: $z^2 + z + 1 = 0$.
3. A second example
In this question, we set: $z = -\dfrac{1}{2} + \mathrm{i}\dfrac{\sqrt{3}}{2}$. a. Determine the exponential form of $z$, then those of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_2$ with affix $z^2$ and $P_2$, with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_2$ and $P_2$ are collinear.
Part B
Let $z$ be a non-zero complex number. We denote by $N$ the point with affix $z^2$ and $P$ the point with affix $\dfrac{1}{z}$.

1. Establish that, for every complex number different from 0, we have: $$z^2 - \frac{1}{z} = \left(z^2 + z + 1\right)\left(1 - \frac{1}{z}\right)$$
2. We recall that if $\vec{U}$ is a non-zero vector and $\vec{V}$ is a vector with affixes respectively $z_{\vec{U}}$ and $z_{\vec{V}}$, the vectors $\vec{U}$ and $\vec{V}$ are collinear if and only if there exists a real number $k$ such that $z_{\vec{V}} = k z_{\vec{U}}$. Deduce that, for $z \neq 0$, the points $\mathrm{A}$, $N$ and $P$ defined above are collinear if and only if $z^2 + z + 1$ is a real number.
3. We set $z = x + \mathrm{i}y$, where $x$ and $y$ denote real numbers. Justify that: $z^2 + z + 1 = x^2 - y^2 + x + 1 + \mathrm{i}(2xy + y)$.
4. a. Determine the set of points $M$ with affix $z \neq 0$ such that the points $\mathrm{A}$, $N$ and $P$ are collinear. b. Trace this set of points on the graph given in the appendix.

The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } ; \vec { u } ; \vec { v } )$. In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any answer without justification earns no points. Statement 1: The equation $z - \mathrm { i } = \mathrm { i } ( z + 1 )$ has solution $\sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$. Statement 2: For every real $x \in ] - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \left[ \right.$, the complex number $1 + \mathrm { e } ^ { 2 \mathrm { i } x }$ has exponential form $2 \cos x \mathrm { e } ^ { - \mathrm { i } x }$.
Statement 3: A point M with affix $z$ such that $| z - \mathrm { i } | = | z + 1 |$ belongs to the line with equation $y = - x$.
Statement 4: The equation $z ^ { 5 } + z - \mathrm { i } + 1 = 0$ has a real solution.

The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any unjustified answer receives no points.
Statement 1: The equation $z - \mathrm{i} = \mathrm{i}(z + 1)$ has solution $\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$.
Statement 2: For all real $x \in ]-\frac{\pi}{2}; \frac{\pi}{2}[$, the complex number $1 + \mathrm{e}^{2\mathrm{i}x}$ has exponential form $2\cos x\, \mathrm{e}^{-\mathrm{i}x}$.
Statement 3: A point M with affix $z$ such that $|z - \mathrm{i}| = |z + 1|$ belongs to the line with equation $y = -x$.
Statement 4: The equation $z^5 + z - \mathrm{i} + 1 = 0$ has a real solution.

Exercise 3 (4 points) -- Common to all candidates
For each of the four 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 earns no points. An absence of an answer is not penalised.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the complex number $c = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and the points S and T with affixes respectively $c ^ { 2 }$ and $\frac { 1 } { c }$.

1. Statement 1: The number $c$ can be written as $c = \frac { 1 } { 4 } ( 1 - \mathrm { i } \sqrt { 3 } )$.
2. Statement 2: For all natural integer $n$, $c ^ { 3 n }$ is a real number.
3. Statement 3: The points $\mathrm { O }$, $\mathrm { S }$ and T are collinear.
4. Statement 4: For all non-zero natural integer $n$, $$| c | + \left| c ^ { 2 } \right| + \ldots + \left| c ^ { n } \right| = 1 - \left( \frac { 1 } { 2 } \right) ^ { n } .$$

Exercise 3

The five questions of this exercise are independent. For each of the following statements, indicate whether it is true or false and justify the answer chosen. An unjustified answer is not taken into account. An absence of an answer is not penalized.

1. In the set $\mathbb{C}$ of complex numbers, we consider the equation $(E): z^2 - 2\sqrt{3}\,z + 4 = 0$. We denote $A$ and $B$ the points of the plane whose affixes are the solutions of $(E)$. We denote O the point with affix 0. Statement 1: The triangle $OAB$ is equilateral.
   
2. We denote $u$ the complex number: $u = \sqrt{3} + \mathrm{i}$ and we denote $\bar{u}$ its conjugate. Statement 2: $u^{2019} + \bar{u}^{2019} = 2^{2019}$
   
3. Let $n$ be a non-zero natural number. We consider the function $f_n$ defined on the interval $[0; +\infty[$ by: $$f_n(x) = x\,\mathrm{e}^{-nx+1}$$ Statement 3: For any natural number $n \geqslant 1$, the function $f_n$ admits a maximum.
   
4. We denote $\mathscr{C}$ the representative curve of the function $f$ defined on $\mathbb{R}$ by: $f(x) = \cos(x)\,\mathrm{e}^{-x}$. Statement 4: The curve $\mathscr{C}$ admits an asymptote at $+\infty$.
   
5. Let $A$ be a strictly positive real number. We consider the algorithm: $$\begin{array}{|l} I \leftarrow 0 \\ \text{While } 2^I \leqslant A \\ \quad I \leftarrow I + 1 \\ \text{End While} \end{array}$$ We assume that the variable $I$ contains the value 15 at the end of execution of this algorithm. Statement 5: $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$

We have two urns $U$ and $V$ containing balls. On each of the balls is written one of the numbers $-1$, $1$, or $2$.
Urn $U$ contains one ball bearing the number 1 and three balls bearing the number $-1$. Urn $V$ contains one ball bearing the number 1 and three balls bearing the number 2. We consider a game in which each round proceeds as follows: first we draw at random a ball from urn $U$, we note $x$ the number written on this ball and then we put it in urn $V$. In a second step, we draw at random a ball from urn $V$ and we note $y$ the number written on this ball. We consider the following events:

• $U _ { 1 }$: ``we draw a ball bearing the number 1 from urn $U$, that is $x = 1$'';
• $U _ { - 1 }$: ``we draw a ball bearing the number $-1$ from urn $U$, that is $x = -1$'';
• $V _ { 2 }$: ``we draw a ball bearing the number 2 from urn $V$, that is $y = 2$'';
• $V _ { 1 }$: ``we draw a ball bearing the number 1 from urn $V$, that is $y = 1$'';
• $V _ { - 1 }$: ``we draw a ball bearing the number $-1$ from urn $V$'', that is $y = -1$''.

1. Copy and complete the probability tree.
2. In this game, with each round we associate the complex number $z = x + \mathrm { i } y$.
   Calculate the probabilities of the following events. The answers will be justified. a. $A$: ``$z = -1 - \mathrm { i }$''; b. $B$: ``$z$ is a solution of the equation $t ^ { 2 } + 2 t + 5 = 0$''; c. $C$: ``In the complex plane with an orthonormal coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$ the point $M$ with affixe $z$ belongs to the disk with center O and radius 2''.
3. During a round, we obtain the number 1 on each of the balls drawn. Show that the complex number $z$ associated with this round satisfies $z ^ { 2020 } = - 2 ^ { 1010 }$.

Exercise 4 — Candidates who have not followed the specialization 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. An absence of an answer is not penalized.

1. Let $\left( u _ { n } \right)$ be the sequence defined by $$u _ { 0 } = 4 \text { and for all natural integer } n , u _ { n + 1 } = - \frac { 2 } { 3 } u _ { n } + 1$$ and let $( \nu _ { n } )$ be the sequence defined by $$\text { for all natural integer } n , v _ { n } = u _ { n } - \frac { 2 } { 3 }$$ Statement 1: The sequence $\left( v _ { n } \right)$ is a geometric sequence.
2. Let $( w _ { n } )$ be the sequence defined by, for all non-zero natural integer $n$, $$w _ { n } = \frac { 3 + \cos ( n ) } { n ^ { 2 } } .$$ Statement 2: The sequence $\left( w _ { n } \right)$ converges to 0.
3. Consider the following algorithm: $$\begin{aligned} & U \leftarrow 5 \\ & N \leftarrow 0 \end{aligned}$$ While $U \leqslant 5000$ $$\begin{aligned} & U \leftarrow 3 \times U - 8 \\ & N \leftarrow N + 1 \end{aligned}$$ End While Statement 3: At the end of execution, the variable $U$ contains the value 5000.
4. We denote $\mathbb { C }$ the set of complex numbers. We consider the equation (E) with unknown $z$ in $\mathbb { C }$ $$( z - \mathrm { i } ) \left( z ^ { 2 } + z \sqrt { 3 } + 1 \right) = 0$$ Statement 4: All solutions of equation (E) have modulus 1.
5. We consider the complex numbers $z _ { n }$ defined by $$z _ { 0 } = 2 \text { and for all natural integer } n , z _ { n + 1 } = 2 \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 2 } } z _ { n } .$$ We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For all natural integer $n$, we denote $M _ { n }$ the point with affixe $z _ { n }$. Statement 5: For all natural integer $n$, the point O is the midpoint of the segment $\left[ M _ { n } M _ { n + 2 } \right]$.

If $b$ is a real number satisfying $b ^ { 4 } + \frac { 1 } { b ^ { 4 } } = 6$, find the value of $\left( b + \frac { i } { b } \right) ^ { 16 }$ where $i = \sqrt { - 1 }$.

It is given that the complex number $i - 3$ is a root of the polynomial $3 x ^ { 4 } + 10 x ^ { 3 } + A x ^ { 2 } + B x - 30$, where $A$ and $B$ are unknown real numbers. Find the other roots.

Find all complex solutions to the equation: $$x^{4} + x^{3} + 2x^{2} + x + 1 = 0.$$

For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number.
(a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$.
(b) There is a unique complex number $z$ such that $P ( z ) = 3$.
(c) If $| z | \neq 1$, then $P ( z )$ is infinite.
(d) $P \left( e ^ { i } \right)$ is infinite.

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.
Statements
(33) If $|z| = 1$, then $f(z)$ must be real. (34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$. (35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$. (36) If $f(z)$ is a real number, then $f(z)$ must be positive.

(a) Find all complex solutions of $z^6 = z + \bar{z}$.
(b) For an integer $n > 1$, how many complex solutions does $z^n = z + \bar{z}$ have?

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

When rolling a die twice, let the outcomes be $m$ and $n$ in order. If the probability that $i ^ { m } \cdot ( - i ) ^ { n } = 1$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $i = \sqrt { - 1 }$ and $p , q$ are coprime natural numbers.) [4 points]

1. Let $i$ be the imaginary unit. Then the complex number $( 1 - i ) ( 1 + 2 i ) =$
(A) $3 + 3 i$
(B) $- 1 + 3 i$
(C) $3 + \mathrm { i }$
(D) $- 1 + i$