Found problems: 563
Prove that there is n rational number $r$ such that $cosr\pi=\frac{3}{5}$
For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$?
Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$
A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$
For integers $a$ and $b$ consider the complex number \[\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a, b)$ such that this complex number is a real number.
Prove that a complex number $ z $ is real and positive if for any nonnegative integers $ n, $ the number
$$ z^{2^n} +\bar{z}^{2^n} $$
is real and positive.
[i]Sorin Rădulescu[/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 $|x_{2012}|?$
Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.
Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$.
Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.
Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.
Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$, and that $|a_r - a_s| >1$ for $r\ne s.$
Show that the series
$$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$
converges.
Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds:
\[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]
Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
Let $$z=\frac{1+i}{\sqrt{2}}.$$ What is $$(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?$$
$\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$
Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of
$$\frac{KM + LN}{AC + BD}$$
.
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]
Consider the complex numbers $x,y,z$ such that
$|x|=|y|=|z|=1$. Define the number
$$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$
$\textbf{(a)}$ Prove that $a$ is a real number.
$\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$.
[i] (Stefan Bălăucă & Vlad Robu)[/i]
Let $\omega=\cos\frac{\pi}{5}+\text{i}\sin\frac{\pi}{5}$, which equation has roots $\omega,\omega^3,\omega^7,\omega^9$?
$\text{(A)}x^4+x^3+x^2+x+1=0\qquad\text{(B)}x^4-x^3+x^2-x+1=0$
$\text{(C)}x^4-x^3-x^2+x+1=0\qquad\text{(D)}x^4+x^3+x^2-x+1=0$
Show that the following implication holds for any two complex numbers $x$ and $y$: if $x+y$, $x^2+y^2$, $x^3+y^3$, $x^4+y^4\in\mathbb Z$, then $x^n+y^n\in\mathbb Z$ for all natural n.
A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
$ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$
$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
$\textbf{(A)}~\text{Rectangle}$
$\textbf{(B)}~\text{Rhombus}$
$\textbf{(C)}~\text{Isosceles Trapezium}$
$\textbf{(D)}~\text{Square}$
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.