Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Verify that $\theta(z) \in ]-\pi, \pi[$ and that $R(z) \in \mathcal{P} = \{Z \in \mathbb{C},\, \operatorname{Re}(Z) > 0\}$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Draw on a figure the circle $\mathcal{C}$ with center $O$ and radius $|z|$ and the points $M$ with affixe $z$ and $B$ with affixe $-|z|$. By considering well-chosen angles, show that $$\theta(z) = \operatorname{Arg}(z) = 2\operatorname{Arg}(z + |z|)$$ where $\operatorname{Arg}(z)$ denotes the principal determination of the argument of the complex number $z$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Determine $[R(z)]^2$, $\theta \circ R(z)$ and $|z|^{1/2}\mathrm{e}^{\mathrm{i}\theta(z)/2}$ as functions of $z$, $R(z)$ and $\theta(z)$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Solve using $R$ the equation $Z^2 = z$, with unknown $Z \in \mathbb{C}$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Deduce that $R$ is a bijection from $\mathbb{C} \setminus \mathbb{R}^{-}$ to $\mathcal{P}$. Specify its inverse bijection.

We use the notation $R$ introduced in part I. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$. We denote $$r = \left|R\left(z^2 - 1\right)\right|, \quad s = \left|z + R\left(z^2 - 1\right)\right|, \quad t = \left|z - R\left(z^2 - 1\right)\right|, \quad h = \max(s,t)$$ We also denote $V_n(z) = U_{n+1}(z,-1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$. Prove that, for all $n \in \mathbb{N}$, $$\left|V_n(z)\right| \leqslant \frac{h^{n+1}}{r}$$

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Let $t \in [0, 2\pi]$ be fixed. Determine two complex numbers $\alpha$ and $\beta$, independent of $t$ and $z$, such that $$\mathrm{N}(x,y,t) = -1 + \frac{\alpha}{1 - ze^{-it}} + \frac{\beta}{1 - \bar{z}e^{it}}$$

We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $a$ and $\lambda$ be non-zero complex numbers. Assume that $\frac { a } { \lambda } \notin \mathcal { V } _ { p }$, which means that either $a = \lambda$ or $\frac { a ^ { p } } { \lambda ^ { p } } \neq 1$. Prove that the complex number $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } a ^ { j }$ is non-zero.

Let $z _ { 1 } , \ldots , z _ { n }$ be complex numbers. Show that if $$\left| z _ { 1 } + \cdots + z _ { n } \right| = \left| z _ { 1 } \right| + \cdots + \left| z _ { n } \right|$$ then the vector $\left( \begin{array} { c } z _ { 1 } \\ \vdots \\ z _ { n } \end{array} \right)$ is collinear with the vector $\left( \begin{array} { c } \left| z _ { 1 } \right| \\ \vdots \\ \left| z _ { n } \right| \end{array} \right)$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. If $z \in \mathbb{C}$ satisfies $|z| = 1$, show that $|A(z)| \leq n$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. We also fix an integer $L \geq 1$. We assume in this question that $a_0 = 1$, and we set, for all $z \in \mathbb{C}$, $$F(z) = \prod_{j=0}^{L-1} A\left(z e^{\frac{2i\pi j}{L}}\right)$$
a. Show that there exists $z_0 \in \mathbb{C}$ such that $|z_0| = 1$ and $|F(z_0)| \geq 1$.
b. Show that $|F(z_0)| \leq n^{L-1} \cdot \sup_{\theta \in \left[-\frac{\pi}{L}, \frac{\pi}{L}\right]} \left|A\left(e^{i\theta}\right)\right|$.

Let $n \in \mathbf { N } ^ { * }$ as well as complex numbers $z _ { 1 } , \ldots , z _ { n } , u _ { 1 } , \ldots , u _ { n }$ all of modulus at most 1. Show that
$$\left| \prod _ { k = 1 } ^ { n } z _ { k } - \prod _ { k = 1 } ^ { n } u _ { k } \right| \leq \sum _ { k = 1 } ^ { n } \left| z _ { k } - u _ { k } \right|$$

a) Show that for all $x, y, z, t \in \mathbb{R}$ we have $$N(xE + yI + zJ + tK) = x^2 + y^2 + z^2 + t^2.$$ b) Show that for all $U \in \mathbb{H}^{\mathrm{im}}$ we have $U^2 = -N(U)E$ and that $$\mathbb{H}^{\mathrm{im}} = \left\{ U \in \mathbb{H} \mid U^2 \in \left]-\infty, 0\right] E \right\}.$$

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Compare the real numbers $- x _ { n , k }$ and $x _ { n , n - k }$.

If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.

Let $a < b < c$ be three real numbers and $w$ denote a complex cube root of unity. If $\left( a + bw + cw ^ { 2 } \right) ^ { 3 } + \left( a + bw ^ { 2 } + cw \right) ^ { 3 } = 0$, then which of the following must be true?
(a) $a + b + c = 0$
(b) $abc = 0$
(c) $ab + bc + ca = 0$
(d) $b = ( c + a ) / 2$.

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(A) $z$ is neither real nor purely imaginary
(B) $z$ is real
(C) $z$ is purely imaginary
(D) none of the above

The set of complex numbers $z$ satisfying the equation $$( 3 + 7i ) z + ( 10 - 2i ) \bar { z } + 100 = 0$$ represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite.

Let $\mathbb{C}$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z , w ) \mid z , w \in \mathbb{C} \text{ and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb{C} , \text{ and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(A) $A = B$
(B) $A \subset B$ and $A \neq B$
(C) $B \subset A$ and $B \neq A$
(D) none of the above

Let $z_1$ be a purely imaginary complex number and $z_2$ be any complex number such that $|z_1 + z_2| = |z_1 - z_2|$. Find the circumcentre of the triangle with vertices $0, z_1, z_2$.
(A) $z_1/2$ (B) $z_2/2$ (C) $(z_1 + z_2)/2$ (D) $(z_1 - z_2)/2$

Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(a) $A = B$
(b) $A \subset B$ and $A \neq B$
(c) $B \subset A$ and $B \neq A$
(d) none of the above.

Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(a) $z$ is neither real nor purely imaginary
(b) $z$ is real
(c) $z$ is purely imaginary
(d) none of the above.

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always