Let $n \in \mathbb{N}, n \ge 3.$ $a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$ $b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$

Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation\[ r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0 \]is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write \[ r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i} \]to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions \[ k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0. \]Find the sum of all such $k$.

Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that \[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]

Given that \[ \frac{\cos(x) \plus{} \cos(y) \plus{} \cos(z)}{\cos(x\plus{}y\plus{}z)} \equal{} \frac{\sin(x)\plus{} \sin(y) \plus{} \sin(z)}{\sin(x \plus{} y \plus{} z)} \equal{} a,\] show that \[ \cos(y\plus{}z) \plus{} \cos(z\plus{}x) \plus{} \cos(x\plus{}y) \equal{} a.\]

Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]

Let $n\ge 2$ be an integer. Let $z_1, z_2,..., z_n$ be complex numbers in such a way that for all integers $k$ with $1\le k\le n$: $$\Pi_{i = 1,i\ne k}^{n} (z_k- z_i) = \Pi_{i = 1,i\ne k}^{n} (z_k+ z_i).$$ Show that two of them are the same.

Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of \[\left|\frac{x + y}{1 + x\overline{y}}\right|\]

What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$

A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule \[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}} \]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 2005\qquad \textbf{(E)}\ 2^{2005}$

In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$?

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$, such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$

Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]

$z_1,z_2$ are complex numbers. $|z_1|=3,|z_2|=5,|z_1+z_2|=7$, then $\arg(\frac{z_2}{z_1})^3=$________.

Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that \[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$. (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation \[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \] has at least one complex root with norm at most $3n|\omega|$.

Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$. Show that for every $a\in\mathbb{C}$ and $r>0$, the number \[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\] is an integer. Interpet this result geometrically. [i]Octavian Stănășilă[/i]

Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n=2$. Then show that the same $k$ works for any $n \geqslant 2$.]

Let $ABC$ be an equilateral triangle. On the small arc $AB{}$ of its circumcircle $\Omega$, consider the point $N{}$ such that the small arc $NB$ measures $30^\circ{}$. The perpendiculars from $N{}$ onto $AC$ and $AB$ intersect $\Omega$ again at $P{}$ and $Q{}$ respectively. Let $H_1,H_2$ and $H_3$ be the orthocenters of the triangles $NAB, QBC$ and $CAP$ respectively. [list=a] [*]Prove that the triangle $NPQ$ is equilateral. [*]Prove that the triangle $H_1H_2H_3$ is equilateral. [/list]

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.

Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and $$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$ for all natural $ k\ge 2. $ [b]a)[/b] Find $ a_2. $ [b]b)[/b] Determine this sequence.

Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$. Define $G(n)$ as the average length of the diagonals of this $n$-gon. Prove that if $n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}$.

Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ [i]Gheorghe Iurea[/i]