Found problems: 563
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.
Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $
[i]C. Merticaru[/i]
Let $a,b,c,x,y,$ and $z$ be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If $xy+yz+xz=67$ and $x+y+z=2010$, find the value of $xyz$.
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
$$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$
Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.
Prove that the sum of the $ \text{2005-th} $ powers of three pairwise distinct complex numbers is the imaginary unit if their modulus are equal and the sum of these numbers is the imaginary unit.
Let $a,b,c$ be some complex numbers. Prove that
$$|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2}$$
if the denominators are not 0
The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?
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 $100x$?
(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that
$$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which
$$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$
Complex numbers $z_1,z_2$ satisfy that $|z_1|=2,|z_2|=3,3z_1-2z_2=\frac{3}{2}-\text{i}$, then $z_1\cdot z_2=$________.
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?
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 \[ \cos(y\plus{}z) \plus{} \cos(z\plus{}x) \plus{} \cos(x\plus{}y) \equal{} a.\]
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.
Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation
\[\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}\]
Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$
Let $P_1P_2P_3\dots P_{12}$ be a regular dodecagon. Show that \[\left|P_1P_2\right|^2 + \left|P_1P_4\right|^2 + \left|P_1P_6\right|^2 + \left|P_1P_8\right|^2 + \left|P_1P_{10}\right|^2 + \left|P_1P_{12}\right|^2\] is equal to \[\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.\]
What is the value of the sum
\[
\sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]
where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
Complex number $z=1-\sin\theta+\text{i}\cos\theta\left(\frac{\pi}{2}<\theta<\pi\right)$, find the range value of $\arg{\overline{z}}$.
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 $P$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have positive imaginary part, and suppose that $P=r(\cos \theta^\circ+i\sin \theta^\circ),$ where $0<r$ and $0\le \theta <360.$ Find $\theta.$
Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
The following sets of points are considered in the plane:
$A=\{$ affixes of complexes $z$ such that arg $(z - (2 + 3i))=\pi /4\}$,
$B =\{$ affixes of complexes $z$ such that mod $( z- (2 + i)<2\}$.
Determine the orthogonal projection on the $X$ axis of $A \cap B$.