Olympiad problems

2000 CentroAmerican, 3

Tags: geometry , parallelogram , trigonometry , analytic geometry , graphing lines , slope , complex number

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$.

2004 Gheorghe Vranceanu, 4

Tags: linear algebra , matrix , complex number

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

2014 Brazil Team Selection Test, 4

Tags: geometry , complex number , hexagon

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.

2001 Croatia National Olympiad, Problem 1

Tags: complex number , algebra

Let $z\ne0$ be a complex number such that $z^8=\overline z$. What are the possible values of $z^{2001}$?

2008 Romania National Olympiad, 2

Tags: inequalities , circumcircle , complex number , algebra , geometry

Let $ a,b,c$ be 3 complex numbers such that \[ a|bc| \plus{} b|ca| \plus{} c|ab| \equal{} 0.\] Prove that \[ |(a\minus{}b)(b\minus{}c)(c\minus{}a)| \geq 3\sqrt 3 |abc|.\]

1991 AMC 12/AHSME, 18

Tags: hyperbola , parabola , complex number , conic

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a $ \textbf{(A)}\text{ right triangle}\qquad\textbf{(B)}\text{ circle}\qquad\textbf{(C)}\text{ hyperbola}\qquad\textbf{(D)}\text{ line}\qquad\textbf{(E)}\text{ parabola} $

2004 Croatia National Olympiad, Problem 1

Tags: complex number , inequality , complex inequality , inequalities

Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

2012 Gheorghe Vranceanu, 1

Tags: complex number , linear algebra , matrix

[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $ [b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b] Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $ [i]Adrian Troie[/i]

2012 China Team Selection Test, 1

Tags: inequalities , complex number , triangle inequality , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

2008 Bulgaria Team Selection Test, 2

Tags: trigonometry , complex number , inequalities , geometry

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

1995 All-Russian Olympiad, 3

Tags: quadratic , polynomial , symmetry , function , complex number , algebra

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

2002 VJIMC, Problem 1

Tags: complex number , system of equations , algebra

Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}

2006 Italy TST, 3

Tags: polynomial , complex number , algebra

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

2006 Italy TST, 3

Tags: polynomial , complex number , algebra

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

2025 Romania National Olympiad, 4

Tags: complex number , algebra , modulus

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

2012 Romanian Masters In Mathematics, 2

Tags: circumcircle , trigonometry , complex number , geometry

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

2022 MMATHS, 11

Tags: complex number , algebra

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

2001 AIME Problems, 14

Tags: trigonometry , complex number , absolute value

There are $2n$ complex numbers that satisfy both $z^{28}-z^{8}-1=0$ and $|z|=1$. These numbers have the form $z_{m}=\cos\theta_{m}+i\sin\theta_{m}$, where $0\leq\theta_{1}<\theta_{2}< \dots <\theta_{2n}<360$ and angles are measured in degrees. Find the value of $\theta_{2}+\theta_{4}+\dots+\theta_{2n}$.

2009 ISI B.Math Entrance Exam, 1

Tags: complex number

2006 Petru Moroșan-Trident, 1

Tags: complex number , algebra

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]

2019 AMC 12/AHSME, 21

Tags: complex number

Let $$z=\frac{1+i}{\sqrt{2}}.$$ What is $$(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?$$ $\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$

2006 Romania Team Selection Test, 1

Tags: circumcircle , geometric transformation , rotation , similar triangles , complex number , geometry

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

1993 National High School Mathematics League, 9

Tags: complex number

If $z\in\mathbb{C},\arg{(z^2-4)}=\frac{5}{6}\pi,\arg{(z^2+4)}=\frac{\pi}{3}$, then the value of $z$ is________.

2008 Grigore Moisil Intercounty, 2

Tags: complex number , polynom , linear algebra

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]

Gheorghe Țițeica 2024, P4

Tags: algebra , complex number

Determine positive integers $n\geq 3$ such that there exists a set $M$ of $n$ complex numbers and a positive integer $m$ such that $(1+z_1z_2z_3)^m=1$ for all pairwise distinct $z_1,z_2,z_3\in M$. [i]Vlad Matei[/i]