Found problems: 563
2000 Romania National Olympiad, 3
A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon.
Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.
2024 Mexican University Math Olympiad, 3
Consider a multiplicative function \( f \) from the positive integers to the unit disk centered at the origin, that is, \( f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} \) such that \( f(mn) = f(m)f(n) \). Prove that for every \( \epsilon > 0 \) and every integer \( k > 0 \), there exist \( k \) distinct positive integers \( a_1, a_2, \dots, a_k \) such that \( \text{gcd}(a_1, a_2, \dots, a_k) = k \) and \( d(f(a_i), f(a_j)) < \epsilon \) for all \( i, j = 1, \dots, k \).
2005 Miklós Schweitzer, 9
prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.
2002 National High School Mathematics League, 7
Complex numbers $|z_1|=2,|z_2|=3$, and the intersection angle between the vectors corresponding to $z_1,z_2$ is $60^{\circ}$, then $\frac{|z_1+z_2|}{|z_1-z_2|}=$________.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
2010 All-Russian Olympiad, 3
Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point.
1976 Spain Mathematical Olympiad, 5
Show that the equation
$$z^4 + 4(i + 1)z + 1 = 0$$
has a root in each quadrant of the complex plane.
2013 BMT Spring, 4
Given a complex number $z$ satisfies $\operatorname{Im}(z)=z^2-z$, find all possible values of $|z|$.
2025 All-Russian Olympiad, 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[
a^2 + b^2 + 1 = 2ab
\]
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[
c^2 + d^2 + 2025 = 2cd
\]
1965 Miklós Schweitzer, 3
Let $ a,b_0,b_1,b_2,...,b_{n\minus{}1}$ be complex numbers, $ A$ a complex square matrix of order $ p$, and $ E$ the unit matrix of order $ p$. Assuming that the eigenvalues of $ A$ are given, determine the eigenvalues of the matrix
\[ B\equal{}\begin{pmatrix} b_0E&b_1A&b_2A^2&\cdots&b_{n\minus{}1}A^{n\minus{}1} \\
ab_{n\minus{}1}A^{n\minus{}1}&b_0E&b_1A&\cdots&b_{n\minus{}2}A^{n\minus{}2}\\
ab_{n\minus{}2}A^{n\minus{}2}&ab_{n\minus{}1}A^{n\minus{}1}&b_0E&\cdots&b_{n\minus{}3}A^{n\minus{}3}\\
\vdots&\vdots&\vdots&\ddots&\vdots&\\
ab_1A&ab_2A^2&ab_3A^3&\cdots&b_0E
\end{pmatrix}\quad\]
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2004 Postal Coaching, 12
Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations.
2019 LIMIT Category A, Problem 9
$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
$\textbf{(A)}~\text{Rectangle}$
$\textbf{(B)}~\text{Rhombus}$
$\textbf{(C)}~\text{Isosceles Trapezium}$
$\textbf{(D)}~\text{Square}$
2004 France Team Selection Test, 3
Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$.
Prove that there exists an equilateral triangle whose vertices belong to distinct disks.
Prove that such a triangle has side-length greater than 96.
1969 Czech and Slovak Olympiad III A, 4
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.
1986 Putnam, B2
Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$.
2012 Romania National Olympiad, 2
[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:
\[3\le |z-a|+|z-b|+|z-c|\le 4,\]
for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]
2011-2012 SDML (High School), 4
What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$
1990 IMO Shortlist, 16
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.
2001 Stanford Mathematics Tournament, 5
What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2022-23 IOQM India, 14
Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
\\
If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$.
PEN S Problems, 6
Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.
2019 Teodor Topan, 2
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]