Found problems: 563
2010 Olympic Revenge, 6
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 $\angle BAX = \angle CAZ$.
2012 Bogdan Stan, 2
Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $
2010 Laurențiu Panaitopol, Tulcea, 3
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]
2014 Ukraine Team Selection Test, 3
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
1989 Spain Mathematical Olympiad, 5
Consider the set $D$ of all complex numbers of the form $a+b\sqrt{-13}$ with $a,b \in Z$. The number $14 = 14+0\sqrt{-13}$ can be written as a product of two elements of $D$: $14 = 2 \cdot 7$. Find all possible ways to express $14$ as a product of two elements of $D$.
2014 Romania National Olympiad, 4
Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then:
[b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$
[b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$
Prove that $ P \Longleftrightarrow Q$
2018 Purple Comet Problems, 14
A complex number $z$ whose real and imaginary parts are integers satis fies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$
.
2010 Contests, 3
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 $|bc|$.
2018 AMC 12/AHSME, 22
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$
$\textbf{(A) } 20 \qquad
\textbf{(B) } 21 \qquad
\textbf{(C) } 22 \qquad
\textbf{(D) } 23 \qquad
\textbf{(E) } 24 $
2001 India IMO Training Camp, 1
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
2000 CentroAmerican, 3
Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.
2010 Contests, 1
Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.
1993 National High School Mathematics League, 7
Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.
2013 IMC, 5
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]
2007 ISI B.Math Entrance Exam, 3
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 $n$.
2019 IFYM, Sozopol, 6
Prove that for $\forall$ $z\in \mathbb{C}$ the following inequality is true:
$|z|^2+2|z-1|\geq 1$,
where $"="$ is reached when $z=1$.
2025 ISI Entrance UGB, 4
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 : $2025 = 3^4 \times 5^2$)
1990 IMO Longlists, 58
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.
2010 AIME Problems, 7
Let $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$, where $ a$, $ b$, and $ c$ are real. There exists a complex number $ w$ such that the three roots of $ P(z)$ are $ w \plus{} 3i$, $ w \plus{} 9i$, and $ 2w \minus{} 4$, where $ i^2 \equal{} \minus{} 1$. Find $ |a \plus{} b \plus{} c|$.
2003 Tournament Of Towns, 5
Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?
1954 Czech and Slovak Olympiad III A, 2
Let $a,b$ complex numbers. Show that if the roots of the equation $z^2+az+b=0$ and 0 form a triangle with the right angle at the origin, then $a^2=2b\neq0.$ Also determine whether the opposite implication holds.
2019 China Team Selection Test, 1
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
2005 Serbia Team Selection Test, 1
Prove that there is n rational number $r$ such that $cosr\pi=\frac{3}{5}$
2021 BMT, 20
For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.
2009 Kazakhstan National Olympiad, 2
Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.
Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively.
Prove, that $I, O, H$ lies on one line.