The objective of this exercise is to find a method to construct a regular pentagon with straightedge and compass. In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ), we consider the regular pentagon $A _ { 0 } A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$, with center $O$ such that $\overrightarrow { O A _ { 0 } } = \vec { u }$. We recall that in the regular pentagon $A _ { 0 } A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$:

• the five sides have the same length;
• the points $A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 }$ and $A _ { 4 }$ belong to the unit circle;
• for any integer $k$ belonging to $\{ 0 ; 1 ; 2 ; 3 \}$ we have $\left( \overrightarrow { O A _ { k } } ; \overrightarrow { O A _ { k + 1 } } \right) = \frac { 2 \pi } { 5 }$.

1. We consider the points $B$ with affix $-1$ and $J$ with affix $\frac { \mathrm { i } } { 2 }$.
   The circle $\mathscr { C }$ with center $J$ and radius $\frac { 1 } { 2 }$ intersects the segment $[ B J ]$ at a point $K$. Calculate $B J$, then deduce $B K$.
2. a. Give in exponential form the affix of point $A _ { 2 }$. Justify briefly. b. Prove that $B A _ { 2 } { } ^ { 2 } = 2 + 2 \cos \left( \frac { 4 \pi } { 5 } \right)$. c. A computer algebra system displays the results below, which may be used without justification:
   

   \multicolumn{2}{|l|}{Formal calculation}
   1	\begin{tabular}{ l } $\cos \left( 4 ^ { * } \mathrm { pi } / 5 \right)$
   $\rightarrow \frac { 1 } { 4 } ( - \sqrt { 5 } - 1 )$

   \hline 2 & $\operatorname { sqrt } ( ( 3 - \operatorname { sqrt } ( 5 ) ) / 2 )$ \hline & $\rightarrow \frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$ \hline \end{tabular}
   ``sqrt'' means ``square root'' Deduce, using these results, that $B A _ { 2 } = B K$.
3. In the coordinate system ( $\mathrm { O} , \vec { u } , \vec { v }$ ) provided in the appendix, construct a regular pentagon with straightedge and compass. Do not use a protractor or the ruler's graduations and leave the construction lines visible.

Let $A , B , C$ be angles such that $e ^ { i A } , e ^ { i B } , e ^ { i C }$ form an equilateral triangle in the complex plane. Find values of the given expressions. a) $e ^ { i A } + e ^ { i B } + e ^ { i C }$
Answer: $\_\_\_\_$ b) $\cos A + \cos B + \cos C$
Answer: $\_\_\_\_$ c) $\cos 2 A + \cos 2 B + \cos 2 C$
Answer: $\_\_\_\_$ d) $\cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C$
Answer: $\_\_\_\_$

(a) Count the number of roots $w$ of the equation $z^{2019} - 1 = 0$ over complex numbers that satisfy $|w + 1| \geq \sqrt{2 + \sqrt{2}}$.
(b) Find all real numbers $x$ that satisfy the following equation: $$\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$$

[7 points] Let $z = e^{\left(\frac{2\pi i}{n}\right)}$. Here $n \geq 2$ is a positive integer, $i^{2} = -1$ and the real number $\frac{2\pi}{n}$ can also be considered as an angle in radians.
(i) Show that $\displaystyle\sum_{k=0}^{n-1} z^{k} = 0$.
(ii) Show that $\displaystyle\sum_{k=0}^{8} \cos(40k+1)^{\circ} = 0$, i.e., $\cos(1^{\circ}) + \cos(41^{\circ}) + \cos(81^{\circ}) + \cos(121^{\circ}) + \cdots + \cos(241^{\circ}) + \cos(281^{\circ}) + \cos(321^{\circ}) = 0$.

Show that $r_1$ and $r_2$ are nonzero and that $r_1/r_2$ belongs to $\mathbb{U}_{n+1}$.

Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that $$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$

Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$, where $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$
Show that there exists $U \in \mathbb{C}_{2n}[X]$ such that, for all $\theta \in \mathbb{R}$, $f(\theta) = \mathrm{e}^{-\mathrm{i}n\theta} U(\mathrm{e}^{\mathrm{i}\theta})$.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1 - \omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$, where $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$ Show that there exists $U \in \mathbb{C}_{2n}[X]$ such that, for all $\theta \in \mathbb{R}$, $f(\theta) = \mathrm{e}^{-\mathrm{i}n\theta} U(\mathrm{e}^{\mathrm{i}\theta})$.

Let $n$ be a non-zero natural number. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Let $f \in \mathcal{S}_n$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1-\omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$

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) $b = ( c + a ) / 2$.

Let $w$ be a primitive cube root of unity. Simplify $\dfrac{1}{z-3} + \dfrac{1}{z-3w} + \dfrac{1}{z-3w^2}$.

Suppose $z \in \mathbb { C }$ is such that the imaginary part of $z$ is non-zero and $z ^ { 25 } = 1$. Then $$\sum _ { k = 0 } ^ { 2023 } z ^ { k }$$ equals
(A) 0.
(B) 1.
(C) $- 1 - z ^ { 24 }$.
(D) $- z ^ { 24 }$.

Let $w = \frac { \sqrt { 3 } + \mathrm { i } } { 2 }$ and $P = \left\{ w ^ { n } : n = 1,2,3 , \ldots \right\}$. Further $H _ { 1 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z > \frac { 1 } { 2 } \right\}$ and $H _ { 2 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z < \frac { - 1 } { 2 } \right\}$, where $\mathbb { C }$ is the set of all complex numbers. If $z _ { 1 } \in P \cap H _ { 1 }$, $z _ { 2 } \in P \cap H _ { 2 }$ and $O$ represents the origin, then $\angle z _ { 1 } O z _ { 2 } =$
(A) $\frac { \pi } { 2 }$
(B) $\frac { \pi } { 6 }$
(C) $\frac { 2 \pi } { 3 }$
(D) $\frac { 5 \pi } { 6 }$

Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots , A _ { 8 }$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A _ { i }$ denote the distance between the points $P$ and $A _ { i }$ for $i = 1,2 , \ldots , 8$. If $P$ varies over the circle, then the maximum value of the product $P A _ { 1 } \cdot P A _ { 2 } \cdots P A _ { 8 }$, is

Let $\omega$ be a complex number such that $2 \omega + 1 = z$ where $z = \sqrt { - 3 }$. If $$\left| \begin{array} { c c c } { 1 } & { 1 } & { 1 } \\ { 1 } & { - \omega ^ { 2 } - 1 } & { \omega ^ { 2 } } \\ { 1 } & { \omega ^ { 2 } } & { \omega ^ { 7 } } \end{array} \right| = 3 k$$ then $k$ is equal to:
(1) $z$
(2) $- z$
(3) $- 1$
(4) 1

The value of $\sum _ { k = 1 } ^ { 10 } \left( \sin \frac { 2 k \pi } { 11 } + i \cos \frac { 2 k \pi } { 11 } \right)$ is:
(1) 1
(2) $- 1$
(3) $- i$
(4) $i$

The value of $\cos \left( \frac { 2 \pi } { 7 } \right) + \cos \left( \frac { 4 \pi } { 7 } \right) + \cos \left( \frac { 6 \pi } { 7 } \right)$ is equal to
(1) $- 1$
(2) $- \frac { 1 } { 2 }$
(3) $- \frac { 1 } { 3 }$
(4) $- \frac { 1 } { 4 }$