Olympiad problems

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.

2000 District Olympiad (Hunedoara), 2

Tags: complex number , algebra

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. $

2011 District Olympiad, 2

Tags: complex number , absolute value , geometry , rectangle , trigonometry

[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle. [b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication: $$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$

2008 China Team Selection Test, 3

Tags: gauss , polynomial , geometry , complex number , combinatorial geometry , algebra

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

1987 AMC 12/AHSME, 28

Tags: algebra , polynomial , complex number

Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ -a \qquad\textbf{(E)}\ -b $

2009 ISI B.Stat Entrance Exam, 7

Tags: geometry , circumcircle , trigonometry , complex number

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $\frac{x}{2\sin 36^\circ}$.

2003 SNSB Admission, 5

Tags: complex analysis , function , complex number , calculus , derivative

Let be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $ Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $

2019 LIMIT Category B, Problem 2

Tags: complex number , algebra

Let $\mathbb C$ denote the set of all complex numbers. Define $$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$ $\textbf{(B)}~A\subset B\text{ and }A\ne B$ $\textbf{(C)}~B\subset A\text{ and }B\ne A$ $\textbf{(D)}~\text{None of the above}$

2004 Nicolae Coculescu, 3

Tags: complex number , algebra

Let be three nonzero complex numbers $ a,b,c $ satisfying $$ |a|=|b|=|c|=\left| \frac{a+b+c-abc}{ab+bc+ca-1} \right| . $$ Prove that these three numbers have all modulus $ 1 $ or there are two distinct numbers among them whose sum is $ 0. $ [i]Costel Anghel[/i]

2004 India IMO Training Camp, 1

Tags: parallelogram , complex number , circumcircle , geometry

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

2006 Macedonia National Olympiad, 4

Tags: circumcircle , trigonometry , complex number , trig identities , law of sines , geometry

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 $\sum_{i=1}^n(-1)^iMA_i=0$.

1995 National High School Mathematics League, 7

Tags: complex number

$\alpha,\beta$ are conjugate complex numbers. If $|\alpha-\beta|=2\sqrt3$, $\frac{\alpha}{\beta^2}$ is a real number, then $|\alpha|=$________.

1990 National High School Mathematics League, 5

Tags: complex number

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

2014 Online Math Open Problems, 29

Tags: geometric transformation , reflection , analytic geometry , complex number , law of cosines

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]

2007 Grigore Moisil Intercounty, 2

Tags: algebra , polynomial , complex number

Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.

1967 Czech and Slovak Olympiad III A, 1

Tags: algebra , equation , complex number

Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

2010 China Team Selection Test, 2

Tags: polynomial , function , complex number , algebra

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

2013 Online Math Open Problems, 41

Tags: quadratic , geometry , trigonometry , function , complex number

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]

1998 National High School Mathematics League, 13

Tags: complex number

Complex number $z=1-\sin\theta+\text{i}\cos\theta\left(\frac{\pi}{2}<\theta<\pi\right)$, find the range value of $\arg{\overline{z}}$.

2004 AMC 12/AHSME, 16

Tags: function , trigonometry , algebra , polynomial , complex number

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2021 SAFEST Olympiad, 5

Tags: algebra , polynomial , complex number

Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$. Remark: Remember that the roots of a polynomial are not necessarily real numbers.

2007 Gheorghe Vranceanu, 2

Tags: complex number , imaginary numbers , algebra , analytic geometry , complex geometry , geometry

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. $

2015 NIMO Problems, 5

Tags: algebra , polynomial , complex number

Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials \[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \quad\text{ and }\quad Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] Let $S$ be the set of all complex numbers which are a root of [i]either[/i] $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$ [i]Proposed by Michael Tang[/i]

2012 AIME Problems, 6

Tags: trigonometry , parabola , complex number , conic

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.

2008 Romania National Olympiad, 2

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

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