Found problems: 563
2007 Nicolae Păun, 2
Prove that the real and imaginary part of the number $ \prod_{j=1}^n (j^3+\sqrt{-1}) $ is positive, for any natural numbers $ n. $
[i]Nicolae Mușuroia[/i]
1991 National High School Mathematics League, 2
$a,b,c$ are three non-zero-complex numbers, and $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$, then the value of $\frac{a+b-c}{a-b+c}$ is ($\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}$)
$\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2$
1992 USAMO, 5
Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]
1969 Spain Mathematical Olympiad, 2
Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .
2013 NIMO Problems, 5
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
PEN A Problems, 44
Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.
2019 Jozsef Wildt International Math Competition, W. 39
Let $u$, $v$, $w$ complex numbers such that: $u + v + w = 1$, $u^2 + v^2 + w^2 = 3$, $uvw = 1$. Prove that
[list=1]
[*] $u$, $v$, $w$ are distinct numbers two by two
[*] If $S(k)= u^k + v^k + w^k$, then $S(k)$ is an odd natural number
[*] The expression$$\frac{u^{2n+1} - v^{2n+1}}{u-v}+\frac{v^{2n+1}-w^{2n+1}}{v-w}+\frac{w^{2n+1}-u^{2n+1}}{w-u}$$is an integer number.
[/list]
2016 Romania National Olympiad, 3
[b]a)[/b] Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $
[b]b)[/b] Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $
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
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\]
2014 NIMO Problems, 5
Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$.
[i]Proposed by Lewis Chen[/i]
2001 National High School Mathematics League, 8
Complex numbers $z_1,z_2$ satisfy that $|z_1|=2,|z_2|=3,3z_1-2z_2=\frac{3}{2}-\text{i}$, then $z_1\cdot z_2=$________.
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 $\sum_{i=1}^n(-1)^iMA_i=0$.
2024 Turkey Olympic Revenge, 2
In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following:
[list]
[*] For marked points $X,Y$, mark the reflection of $X$ across $Y$.
[*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$.
[*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$.
[/list]
No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves,
a) The circumcenter of $\triangle ABC$.
b) The incenter of $\triangle ABC$.
Proposed by [i]Deniz Can Karaçelebi[/i]
1998 National High School Mathematics League, 13
Complex number $z=1-\sin\theta+\text{i}\cos\theta\left(\frac{\pi}{2}<\theta<\pi\right)$, find the range value of $\arg{\overline{z}}$.
2007 Grigore Moisil Intercounty, 2
Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.
2021 China National Olympiad, 1
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$.
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.
1996 ITAMO, 5
Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.
1949 Miklós Schweitzer, 7
Find the complex numbers $ z$ for which the series
\[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\]
converges and find its sum.
2010 India IMO Training Camp, 10
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
2014 NIMO Problems, 8
Let $a$, $b$, $c$, $d$ be complex numbers satisfying
\begin{align*}
5 &= a+b+c+d \\
125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$.
[i]Proposed by Evan Chen[/i]
2019 ISI Entrance Examination, 3
Let $\Omega=\{z=x+iy~\in\mathbb{C}~:~|y|\leqslant 1\}$. If $f(z)=z^2+2$, then draw a sketch of $$f\Big(\Omega\Big)=\{f(z):z\in\Omega\}$$ Justify your answer.
1983 IMO Longlists, 53
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 $k.$
2013 Online Math Open Problems, 41
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
[hide="Clarifications"]
[list]
[*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
[*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide]
[i]Ray Li[/i]
2005 QEDMO 1st, 8 (Z2)
Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.