Prove that the real and imaginary part of the number $ \prod_{j=1}^n (j^3+\sqrt{-1}) $ is positive, for any natural numbers $ n. $ [i]Nicolae Mușuroia[/i]

Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$

Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

[b]a)[/b] Let $ A $ denote the complex numbers of modulus $ 1/3, $ and $ B $ denote the complex numbers of modulus at least $ 1/2. $ Show that $ A+B=AB\neq\mathbb{C} . $ [b]b)[/b] Prove that there is no family $ Y $ of complex numbers that satisfies $ X+Y=XY\neq\mathbb{C} , $ where $ X $ denotes the complex numbers of modulus $ 1. $

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

Determine a polynomial of non-negative real coefficients that satisfies the following two conditions: $$p(0) = 0, p(|z|) \le x^4 + y^4,$$ being $|z|$ the module of the complex number $z = x + iy$ .

Let $n\ge 2$ be a natural number. Find all complex numbers $z$ which simultaneously satisfy the relations $\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;$ $\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.$

There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.

Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $

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

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$. Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$. [i]Proposed by N. Beluhov[/i]

Let $Z=\{ z_1,\ldots, z_{n-1}\}$, $n\ge 2$, be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$, depending on $Z$, such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$, whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$. b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$, whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$, it also holds that $|x_0|>\frac{c_n}{\varepsilon}$. (translated by L. Erdős)

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]

How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that $$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$ then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.

Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

Let F be the midpoint of side BC or triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E respectively. Show that DEF is an isosceles right triangle.

Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$

Determine positive integers $n\geq 3$ such that there exists a set $M$ of $n$ complex numbers and a positive integer $m$ such that $(1+z_1z_2z_3)^m=1$ for all pairwise distinct $z_1,z_2,z_3\in M$. [i]Vlad Matei[/i]

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

For a complex number $ z\equal{}x\plus{}iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that: $ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and $ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$. If $ \alpha\equal{}p\plus{}iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2\plus{}q^2 \le 1$ and $ p\plus{}q \tan \frac{\pi}{5} \le 1$.