Found problems: 563
1989 IMO Longlists, 58
A regular $ n\minus{}$gon $ A_1A_2A_3 \cdots A_k \cdots A_n$ inscribed in a circle of radius $ R$ is given. If $ S$ is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]
1995 Flanders Math Olympiad, 4
Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$.
Define $G(n)$ as the average length of the diagonals of this $n$-gon.
Prove that if $n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}$.
1991 French Mathematical Olympiad, Problem 5
(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
$$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$
1998 IMO Shortlist, 6
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]
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.
2012 Bogdan Stan, 2
Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $
1996 AIME Problems, 11
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 $\theta.$
2000 District Olympiad (Hunedoara), 2
Let $ z_1,z_2,z_3\in\mathbb{C} $ such that
$\text{(i)}\quad \left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1$
$\text{(ii)}\quad z_1+z_2+z_3\neq 0 $
$\text{(iii)}\quad z_1^2 +z_2^2+z_3^2 =0. $
Show that $ \left| z_1^3+z_2^3+z_3^3\right| = 1. $
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.
2019 AMC 12/AHSME, 17
How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$
1971 Spain Mathematical Olympiad, 5
Prove that whatever the complex number $z$ is, it is true that
$$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$
Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
2013 AMC 12/AHSME, 25
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $
2005 Dutch Mathematical Olympiad, 2
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 \[\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.\]
1980 VTRMC, 8
Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$
(a) Find two such complex numbers.
(b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$
(c) Show that $|z^2n -1|$ is rational for every $n.$
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
2022 District Olympiad, P3
A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$
2022 District Olympiad, P2
Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\geq\sqrt{2}$ for all $i\neq j\in\{1,2,3\}.$ Prove that \[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.\][i]Mathematical Gazette[/i]
2013 AIME Problems, 12
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
2008 Grigore Moisil Intercounty, 2
Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that
$$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $
[i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]
2016 District Olympiad, 2
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$$ a^2+b^2+c^2-ab-bc-ca=0. $$
Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.
2022 Nigerian Senior MO Round 2, Problem 3
In triangle $ABC$, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD, DE $ and $EC$ are $1, 3 $ and $5$ respectively. Find the length of $AC$.
1998 Korea Junior Math Olympiad, 5
Regular $2n$-gon is inscribed in the unit circle. Find the sum of the squares of all sides and diagonal lengths in the $2n$-gon.
2013 Romania National Olympiad, 2
To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent:
i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$;
ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$
2010 Postal Coaching, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2004 Gheorghe Vranceanu, 4
Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $