We propose to show by contradiction the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$ We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that: $$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote $$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$ By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. For all $n \in \mathbb{N}$, we define $$\gamma_{n} : \begin{cases} [0,1] \rightarrow \mathbb{C} \\ t \mapsto (2n+1)\pi\, \mathrm{e}^{2\mathrm{i}\pi t} \end{cases}$$ and for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$, $Q_{n} \in \mathcal{E}$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Show that $$\forall n \in \mathbb{N}^{*},\, \forall z \in \mathbb{C}, \quad Q_{n}(z+1) - Q_{n}(z) = n z^{n-1}.$$

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Using the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right),$$ show that there exist two constants $a, b \in \mathbb{R}_{+}^{*}$ such that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$, $$|Q_{n}(z)| \leqslant a\,\mathrm{e}^{bn|z|}.$$

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) $\frac { 5 \pi } { 6 }$

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4} - |z|^{4} = 4iz^{2}$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_{1} - z_{2}|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}(z_{1}) > 0$ and $\operatorname{Re}(z_{2}) < 0$, is $\_\_\_\_$

Let $S$ be the set of all complex numbers $z$ satisfying $\left| z ^ { 2 } + z + 1 \right| = 1$. Then which of the following statements is/are TRUE?
(A) $\left| z + \frac { 1 } { 2 } \right| \leq \frac { 1 } { 2 }$ for all $z \in S$
(B) $| z | \leq 2$ for all $z \in S$
(C) $\left| z + \frac { 1 } { 2 } \right| \geq \frac { 1 } { 2 }$ for all $z \in S$
(D) The set $S$ has exactly four elements

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

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument of $z$, with $- \pi < \arg ( z ) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg ( \omega ) < \pi$. Let
$$\alpha = \arg \left( \sum _ { n = 1 } ^ { 2025 } ( - \omega ) ^ { n } \right) .$$
Then the value of $\frac { 3 \alpha } { \pi }$ 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 greatest value of $c \in R$ for which the system of linear equations $x - cy - cz = 0$, $cx - y + cz = 0$, $cx + cy - z = 0$ has a non-trivial solution, is
(1) $-1$
(2) $2$
(3) $\frac{1}{2}$
(4) $0$

If $z$ and $\omega$ are two complex numbers such that $| z \omega | = 1$ and $\arg ( z ) - \arg ( \omega ) = \frac { 3 \pi } { 2 }$, then $\arg \left( \frac { 1 - 2 \bar { z } \omega } { 1 + 3 \bar { z } \omega } \right)$ is: (Here $\arg ( z )$ denotes the principal argument of complex number $z$)
(1) $\frac { \pi } { 4 }$
(2) $- \frac { 3 \pi } { 4 }$
(3) $- \frac { \pi } { 4 }$
(4) $\frac { 3 \pi } { 4 }$

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) $- \frac { 1 } { 4 }$

The value of $\left( \frac { 1 + \sin \frac { 2 \pi } { 9 } + i \cos \frac { 2 \pi } { 9 } } { 1 + \sin \frac { 2 \pi } { 9 } - i \cos \frac { 2 \pi } { 9 } } \right) ^ { 3 }$ is
(1) $\frac { - 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(2) $\frac { 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(3) $\frac { - 1 } { 2 } ( \sqrt { 3 } - i )$
(4) $\frac { 1 } { 2 } ( \sqrt { 3 } + i )$

The complex number $z = \frac{i-1}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}$ is equal to:
(1) $\sqrt{2}i\left(\cos\frac{5\pi}{12} - i\sin\frac{5\pi}{12}\right)$
(2) $\cos\frac{\pi}{12} - i\sin\frac{\pi}{12}$
(3) $\sqrt{2}\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right)$
(4) $\sqrt{2}\left(\cos\frac{5\pi}{12} + i\sin\frac{5\pi}{12}\right)$

Let $A = \left\{ \theta \in ( 0,2 \pi ) : \frac { 1 + 2 i \sin \theta } { 1 - i \sin \theta } \right.$ is purely imaginary $\}$ Then the sum of the elements in $A$ is
(1) $4 \pi$
(2) $3 \pi$
(3) $\pi$
(4) $2 \pi$

Let $S = \{ z \in \mathbb { C } : \bar { z } = i z ^ { 2 } + \operatorname { Re } ( \bar { z } ) \}$. Then $\sum _ { z \in S } | z | ^ { 2 }$ is equal to
(1) $\frac { 5 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3

If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to
(1) 441
(2) 398
(3) 312
(4) 409

Let us consider the solutions to the equation in the complex number $z$
$$z ^ { 4 } = - 324 \quad \cdots (1)$$
and the solutions to the equation in the complex number $z$
$$z ^ { 4 } = t ^ { 4 } \quad \cdots (2)$$
where $t$ is a positive real number.
(1) To find the solutions to (1), let us set
$$z = r ( \cos \theta + i \sin \theta ) \quad ( r > 0,0 < \theta \leqq 2 \pi )$$
Then
$$z ^ { 4 } = r ^ { \mathbf { M } } ( \cos \mathbf { N } \theta + i \sin \mathbf { N } \theta ) .$$
The values of $r$ and $\theta$ such that this expression is equal to $-324$ are
$$\begin{aligned} & r = \mathbf { O } \sqrt { \mathbf { P } } , \\ & \theta = \frac { \mathbf { Q } } { \mathbf { R } } \pi , \frac { \mathbf { S } } { \mathbf { R } } \pi , \frac { \mathbf { T } } { \mathbf { R } } \pi , \frac { \mathbf { U } } { \mathbf { R } } \pi , \end{aligned}$$
where $\mathbf { Q } < \mathbf { S } < \mathbf { T } < \mathbf { U }$.
(2) There are $\mathbf { V }$ solutions to equation (2), and these solutions are dependent on $t$. Now, consider any one of the solutions to (1) and any one of the solutions to (2), and let $d$ be the distance on the complex number plane between these two solutions. Then, over the interval $0 < t \leqq 4$, the minimum value of $d$ is $\mathbf { W }$ and the maximum value is $\sqrt { \mathbf { X Y } }$.