Found problems: 563
2019 District Olympiad, 2
Let $n \in \mathbb{N}, n \ge 3.$
$a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$
$b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$
2022 Romania National Olympiad, P2
Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\][i]Vlad Cerbu and Sorin Rădulescu[/i]
1985 AIME Problems, 3
Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.
2014 District Olympiad, 1
Solve for $z\in \mathbb{C}$ the equation :
\[ |z-|z+1||=|z+|z-1|| \]
1992 Spain Mathematical Olympiad, 6
For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying
$(x+iy)^n+(x-iy)^n = 2x^n$ .
(a) Determine $S(n)$ for $n = 2,3,4$.
(b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.
2018 District Olympiad, 4
Let $n\ge 2$ be a natural number. Find all complex numbers $z$ which simultaneously satisfy the relations
$\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;$
$\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.$
1995 National High School Mathematics League, 7
$\alpha,\beta$ are conjugate complex numbers. If $|\alpha-\beta|=2\sqrt3$, $\frac{\alpha}{\beta^2}$ is a real number, then $|\alpha|=$________.
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$
1980 IMO, 23
Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\]
2003 Gheorghe Vranceanu, 3
Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $
1985 Spain Mathematical Olympiad, 5
Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.
2004 Mediterranean Mathematics Olympiad, 4
Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and
\[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\]
If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.
1992 AMC 12/AHSME, 28
Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is
$ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $
2014 Online Math Open Problems, 29
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
[i]Proposed by Evan Chen[/i]
2008 iTest Tournament of Champions, 3
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.
2005 USAMTS Problems, 3
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate
and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.
2007 Gheorghe Vranceanu, 2
Let be a natural number $ n\ge 2 $ and an imaginary number $ z $ having the property that $ |z-1|=|z+1|\cdot\sqrt[n]{2} . $ Denote with $ A,B,C $ the points in the Euclidean plane whose representation in the complex plane are the affixes of $
z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} , $ respectively. Prove that $ AB $ is perpendicular to $ AC. $
2021 239 Open Mathematical Olympiad, 5
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
2012 EGMO, 8
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]
1983 AIME Problems, 5
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?
2004 Romania Team Selection Test, 16
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$.
Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]
2017 Harvard-MIT Mathematics Tournament, 16
Let $a$ and $b$ be complex numbers satisfying the two equations
\begin{align*}
a^3 - 3ab^2 & = 36 \\
b^3 - 3ba^2 & = 28i.
\end{align*}
Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a| = M$.
2006 Macedonia National Olympiad, 4
Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ .
$(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$.
$(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.
1990 National High School Mathematics League, 5
Two non-zero-complex numbers $x,y$, satisfy that $x^2+xy+y^2=0$. Then the value of $(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}$ is
$\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}$none above
2019 Jozsef Wildt International Math Competition, W. 13
Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of
\begin{tabular}{|ccc|}
$b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\
$n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\
$n(b - a)$ & $n^2(a - b)$ & $a + n^3b$
\end{tabular}