Found problems: 563
Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies
$$|a^{2021}-N|<2^{-1000}.$$
[i] (Vlad Robu) [/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.
Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime?
[i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]
Find all complex roots (with multiplicities) of the polynomial
$$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[
a^2 + b^2 + 1 = 2ab
\]
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[
c^2 + d^2 + 2025 = 2cd
\]
Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$
My solution was nearly complete...
Show that if $a,b,c$ are complex numbers that such that
\[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\]
then $a,b,c$ are real numbers.
A [i]word[/i] is a finite sequence of letters from some alphabet. A word is [i]repetitive[/i] if it is a concatenation of at least two identical subwords (for example, $ababab$ and $abcabc$ are repetitive, but $ababa$ and $aabb$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
[i]Romania (Dan Schwarz)[/i]
Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.
Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.
Without using tables, find the exact value of the product:
\[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]
Consider any complex units $Z,W$ with $\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.$ Determine and draw the locus of all possible sums $S=Z+W$ in the complex plane.
The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.
Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$.
Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.
If $\theta=\arctan \frac{5}{12}$, $z=\frac{\cos 2\theta+\text{i}\sin2\theta}{239+\text{i}}$, then $\arg z=$________.
For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$.
[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $
[b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b]
Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $
[i]Adrian Troie[/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?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
If $\xi_1,\ldots,\xi_n$ denote the $n$-th roots of unity, evaluate
$$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$
Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$.
[i](United Kingdom) David Monk[/i]
Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?
$ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $
Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let
$$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$
(1) If $n = 6063$, find the maximum value of $S$.
(2) If $n= 2021$, find the maximum value of $S$.
Let $k$ be a real number such that the system \begin{align*} &|25+20i-z|=5\\ &|z-4-k|=|z-3i-k| \\ \end{align*} has exactly one complex solution $z.$ The sum of all possible values of $k$ can be written as $\dfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Here $i=\sqrt{-1}.$
Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.
Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.