In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Let $ a,b_0,b_1,b_2,...,b_{n\minus{}1}$ be complex numbers, $ A$ a complex square matrix of order $ p$, and $ E$ the unit matrix of order $ p$. Assuming that the eigenvalues of $ A$ are given, determine the eigenvalues of the matrix \[ B\equal{}\begin{pmatrix} b_0E&b_1A&b_2A^2&\cdots&b_{n\minus{}1}A^{n\minus{}1} \\ ab_{n\minus{}1}A^{n\minus{}1}&b_0E&b_1A&\cdots&b_{n\minus{}2}A^{n\minus{}2}\\ ab_{n\minus{}2}A^{n\minus{}2}&ab_{n\minus{}1}A^{n\minus{}1}&b_0E&\cdots&b_{n\minus{}3}A^{n\minus{}3}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\\ ab_1A&ab_2A^2&ab_3A^3&\cdots&b_0E \end{pmatrix}\quad\]

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 $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the [i]period[/i] of $z$. Determine the total number of points in $S^1$ of period $2025$. (Hint : $2025 = 3^4 \times 5^2$)

Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.

Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $ [b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that $$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$ if $ \triangle $ is equilateral. [b]b)[/b] Show that $ \triangle $ is equilateral if $$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$

Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.

During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.

Let be a complex number $ z $ having the property that $ \Re \left( z^n \right) >\Im \left( z^n \right) , $ for any natural numbers $ n. $ Show that $ z $ is a positive real number. [i]Laurențiu Panaitopol[/i]

$a_1,a_2,a_3,a_4,a_5$ are non-zero complex numbers, satisfying: $\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases}$ Where $S$ is a real number that $|S|\leq2$ Prove that points that $a_1,a_2,a_3,a_4,a_5$ refers to in the complex plane are concyclic.

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$ $$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$ $$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

What is the largest amount of complex $ z $ solutions a system can have? $ | z-1 || z + 1 | = 1 $ $ Im (z) = b? $ (where $ b $ is a real constant)

Let $ S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) $, where the $ b_j$ are arbitrary and the $ z_j$ are nonzero complex numbers . Prove that \[ |S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.\] [i]G. Halasz[/i]

$z=x+i y$ where $x$ and $y$ are two real numbers. Find the locus of the point $(x, y)$ in the plane, for which $\frac{z+i}{z-i}$ is purely imaginary (that is, it is of the form $i b$ where $b$ is a real number). [Here, $i=\sqrt{-1}$ [list=1] [*] A straight line [*] A circle [*] A parabole [*] None of these [/list]

[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination? [b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$. [b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and $$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

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]

For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.

What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$