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: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$ We now define a sequence $(f_n)_{n\in\mathbf{N}}$ of elements of $\mathcal{E}$ by choosing $f_0$ affine (i.e. $f_0(x) = -1 + 2x$) and $f_{n+1} = Tf_n$ for every natural number $n$.
a) Prove that the sequence $(f_n)$ converges uniformly on $[0,1]$ to a function $f\in\mathcal{E}$.
b) Prove that $Tf = f$.
c) Prove that, for all $x\in[0,1]$, $f(x) = -\overline{f(1-x)}$ and give a geometric interpretation of this relation.

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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$, i.e. $f(x) = \phi_0(f(2x))$ for $x\in[0,\frac{1}{2}]$ and $f(x) = \phi_1(f(2x-1))$ for $x\in]\frac{1}{2},1]$.
Let $(r_n)_{n\geq 1}\in\{0,1\}^{\mathbf{N}^*}$.
a) Show that the series with general term $\frac{r_n}{2^n}$ converges and that its sum $x$ belongs to $[0,1]$.
b) By setting for every natural number $p$, $x_p = \sum_{n=1}^{\infty}\frac{r_{n+p}}{2^n}$, prove the relation: $$f(x) = \phi_{r_1}\circ\phi_{r_2}\circ\ldots\phi_{r_p}\left(f\left(x_p\right)\right)$$ for every non-zero natural number $p$.

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$.
Conversely, let $x\in[0,1[$.
a) Establish that, for every non-zero natural number $n$, $r_n(x)\in\{0,1\}$.
b) Show that, for every non-zero natural number $N$ and every real $x\in[0,1[$: $$\frac{[2^N x]}{2^N} = \sum_{n=1}^{N}\frac{r_n(x)}{2^n} \quad \text{then} \quad x = \sum_{n=1}^{\infty}\frac{r_n(x)}{2^n}.$$
c) Show that if, moreover, $x\in\mathbf{Z}\left[\frac{1}{2}\right]$ then there exists $N\in\mathbf{N}$ such that $r_n(x) = 0$ for every natural number $n > N$.
d) Calculate $f\left(\frac{1}{2}\right)$ and $f\left(\frac{1}{4}\right)$. Recognize $\phi_0\circ\phi_0$ and deduce $f\left(\frac{1}{2^k}\right)$ for all $k\in\mathbf{N}$.

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\mathbf{C}$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$.
a) Show that $f\left([0,1]\cap\mathbf{Z}\left[\frac{1}{2}\right]\right)\subset\tau$.
b) Show that $f([0,1])\subset\tau$.

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, $\tau_0 = \widehat{0\,(-1)\,(-\mathrm{i})}$, $\tau_1 = \widehat{0\,1\,\mathrm{i}}$. 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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\tau$.
Conversely, let $z\in\tau$.
a) Show that we can define two sequences $(z_n)_{n\geq 0}$ and $(r_n)_{n\geq 1}$ in the following way:

• $z_0 = z$ and, if $n\geq 1$:
• if $z_{n-1}\in\tau_0$ then $r_n = 0$ and $z_n = (\phi_0)^{-1}(z_{n-1})$
• otherwise $r_n = 1$ and $z_n = (\phi_1)^{-1}(z_{n-1})$.

Prove that, for every integer $n\in\mathbb{N}$, $z_n$ belongs to $\tau$.
b) Prove that $f\left(\sum_{n=1}^{\infty}\frac{r_n}{2^n}\right) = z$ (one may express $z$ in terms of $z_n$ and the $\phi_{r_i}$).
c) Write a function that takes as argument a complex number $z$ (which we will assume is in $\tau$) and a real number $\epsilon$ and which returns an approximate value to within $\epsilon$ of a preimage of $z$.

The map $f\in\mathcal{E}$ satisfies $Tf = f$, $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, and $f(x) = -\overline{f(1-x)}$ for all $x\in[0,1]$.
a) Prove that $f$ is not injective (one may use the relation $f(1-x) = -\overline{f(x)}$).
b) More generally show that there exists no continuous bijection from $[0,1]$ onto $\tau$ (one may use an argument of arc-connectedness).

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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$.
a) For $(i,j)\in\{0,1\}^2$, determine the complex expression of $\phi_i\circ\phi_j$, recognize it, specify its fixed point and the image of $\tau$. Make a drawing.
b) Let $r_1, r_2, \ldots, r_p$ be elements of $\{0,1\}$. Prove that $\phi = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_p}$ has a unique fixed point which we will not necessarily try to express simply.
c) Exhibit, using the map $f$, a fixed point of $\phi$.
d) Show that the set $X$ of complex numbers $z$ which are fixed points of the composition of a finite number of maps $\phi_0$ and $\phi_1$ is dense in $\tau$.

The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$. Suppose that $f$ is differentiable on $[0,1]$.
Let $x\in[0,1]$, $(\alpha_n)_{n\geq 1}$ and $(\beta_n)_{n\geq 1}$ be two sequences of elements of $[0,1]$, convergent to $x$ and such that $\alpha_n \leq x \leq \beta_n$ and $\alpha_n < \beta_n$ for all $n$.
Show that the sequence with general term $\frac{f(\beta_n) - f(\alpha_n)}{\beta_n - \alpha_n}$ converges to $f'(x)$.

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$.
Let $x\in[0,1]$.
a) If $x\in[0,1[$, by choosing: $$\alpha_n = \frac{r_1(x)}{2} + \cdots + \frac{r_n(x)}{2^n} \text{ and } \beta_n = \alpha_n + \sum_{k=n+1}^{\infty}\frac{1}{2^k},$$ prove that $f$ is not differentiable at $x$.
b) Prove that $f$ is not differentiable at $1$.

We recall that $\mathscr{C}$ was defined as the image of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
In this question, we seek a complex parametrization of $\mathscr{C}$, of the form $$z : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)}$$ where $\rho$ and $\theta$ are two continuous functions from $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ to $\mathbb{R}$, the function $\rho$ taking strictly positive values.
III.B.1) Calculate $\rho(t)$ for all $t \in \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. III.B.2) Represent on the calculator the parametrized arc $$\mathscr{G} : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}t}$$ and reproduce the curve roughly on the paper. What letter does this curve evoke? III.B.3) From the expression of $\gamma(t)$, calculate $\tan\theta(t)$. III.B.4) a) Represent the function $t \mapsto \arctan(2\tan t)$ on the part of the interval $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ on which this function is defined. b) Modify this function to determine the continuous function $\theta$ sought. The result will be verified by representing with the aid of the calculator the parametrized curve $z$. III.B.5) Indicate a sequence of Maple or Mathematica instructions allowing one to obtain this plot.

We define the applications: $$\alpha : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \frac{\pi}{4} + \frac{3\pi}{2n} \mathrm{E}\left(\frac{2n}{3\pi}\left(t - \frac{\pi}{4}\right)\right)$$ $$\omega : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \cos^2\left(\frac{2n}{3}\left(t - \frac{\pi}{4}\right)\right)$$ where $\mathrm{E}(x)$ denotes the integer part of the real number $x$.
III.C.1) Briefly study $\alpha$ and $\omega$, then represent on the same graph the two functions $t \mapsto \alpha(10, t)$ and $t \mapsto \omega(10, t)$. III.C.2) Represent the function $\psi : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, t \mapsto \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)$. III.C.3) We define the function: $$w : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, (n, t) \mapsto \rho(t)\left(1 + \psi(t)\omega(n, t)\right) \mathrm{e}^{\mathrm{i}\theta(\alpha(n, t))}$$ Identify which of the four graphics represents the function $t \mapsto w(40, t)$, and explain why. III.C.4) Write a sequence of Maple or Mathematica instructions allowing one to create the sequence of the first 100 curves (one may create an animation).

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 $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. We work in the orthonormal frame $\mathcal{R}'$ with center $O'$ with affixe $a$, obtained from $\mathcal{R}$ by translation. Show that an equation of the curve $C_a$ in ``polar coordinates $(\rho, \theta)$'' in the frame $\mathcal{R}'$ is $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that $\Omega_z$ is a bounded subset of the plane. Is it open? closed? compact?

Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that the origin $O$ is an interior point of $\Omega_z$.

Let $z \in \mathbb{C}$. Determine the domain of definition $D_z$ of the function $Z \mapsto \dfrac{1}{1 - 2zZ + Z^2}$.

Let $w \in \mathbb{C}$ be a number that is neither real nor purely imaginary.
(a) Show that the equation $$\left|\frac{z-1}{z+1}\right| = \left|\frac{w-1}{w+1}\right|$$ defines a circle in the complex plane, which passes through $w$. Verify that the interval $]-1,1[$ intersects this circle at a unique point; we denote this point by $y$. We will express $y$ in terms of the number $$\lambda = \left|\frac{w-1}{w+1}\right|.$$
(b) Show the inequality $$\left|\frac{1-w}{1-y}\right| > 1.$$
(c) Show that the equation $$\left|\frac{z-w}{z-y}\right| = \left|\frac{1-w}{1-y}\right|$$ defines a circle in the complex plane, which passes through $1$ and through $-1$.
Deduce that, for all $x \in [-1,1] \setminus \{y\}$, we have $$\left|\frac{w-x}{y-x}\right| \geqslant \left|\frac{w-1}{y-1}\right| = \left|\frac{w+1}{y+1}\right|$$

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, $\arg(w)$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that $w = |w|(\cos\theta + i\sin\theta)$. Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations $\arg(z) > \arg(z+1)$ and $\arg(z) > \arg(z+i)$. Then $\cos(\arg(z))$ can take
(a) Any value in the set $(-1/2, 0) \cup (0, 1/2)$ but none from outside
(b) Any value in the interval $(-1, 0)$ but none from outside
(c) Any value in the interval $(0, 1)$ but none from outside
(d) Any value in the set $(-1, 0) \cup (0, 1)$ but none from outside.

Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]
(a) $\operatorname { Re } ( w ) > 0$
(b) $\operatorname { Im } ( w ) \geq 0$
(c) $| w | \leq 1$
(d) $| w | \geq 1$

Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.

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 $z_1$ 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) $(z_1 - z_2)/2$

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

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ 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 $z$ be a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary. Then the minimum value of $|z - (2 + 2i)|$ is
(A) $2\sqrt{2}$
(B) $\sqrt{2}$
(C) $\frac{3}{\sqrt{2}}$
(D) $\frac{1}{\sqrt{2}}$.