Olympiad problems

2020 Miklós Schweitzer, 9

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

2016 District Olympiad, 3

Tags: complex number , algebra

Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression $$ |\alpha x +\beta y| +|\alpha x-\beta y| $$ in each of the following cases: [b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $ [b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $

2000 Romania National Olympiad, 2

Tags: complex number , algebra

Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$ then $ z_1^6=z_2^6 $

2006 Italy TST, 3

Tags: polynomial , algebra , complex number

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.

2010 China Team Selection Test, 2

Tags: function , polynomial , algebra , complex number

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 Kosovo National Mathematical Olympiad, 1

Tags: complex number

Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$

2009 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt , complex number , logarithm

Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

2024 Vietnam Team Selection Test, 5

Tags: complex number , poles and polars , pascal theorem , geometry

Let incircle $(I)$ of triangle $ABC$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $(O)$ be the circumcircle of $ABC$. Ray $EF$ meets $(O)$ at $M$. Tangents at $M$ and $A$ of $(O)$ meet at $S$. Tangents at $B$ and $C$ of $(O)$ meet at $T$. Line $TI$ meets $OA$ at $J$. Prove that $\angle ASJ=\angle IST$.

2011 ELMO Shortlist, 3

Tags: induction , complex number , imaginary numbers , algebra

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

1997 National High School Mathematics League, 15

Tags: complex number

$a_1,a_2,a_3,a_4,a_5$ are non-zero complex numbers, satisfying: $\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases}$ Where $S$ is a real number that $|S|\leq2$ Prove that points that $a_1,a_2,a_3,a_4,a_5$ refers to in the complex plane are concyclic.

2024 Mexican University Math Olympiad, 3

Tags: multiplicative function , greatest common divisor , euclidean distance , complex number , number theory

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 $.

1977 Spain Mathematical Olympiad, 8

Tags: complex number , equilateral , geometry

Determine a necessary and sufficient condition for the affixes of three complex numbers $z_1$ , $z_2$ and $z_3$ are the vertices of an equilateral triangle.

2014 Math Prize For Girls Problems, 16

Tags: geometry , vector , trigonometry , complex number

If $\sin x + \sin y = \frac{96}{65}$ and $\cos x + \cos y = \frac{72}{65}$, then what is the value of $\tan x + \tan y$?

2012 Romania National Olympiad, 2

Tags: inequalities , complex number , geometry , circumcircle

[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]

2013 Iran MO (3rd Round), 5

Tags: polynomial , algebra , complex number , trigonometry

Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon. (30 points)

2017 District Olympiad, 4

Tags: algebra , complex number

Let $ C $ denote the complex unit circle centered at the origin. [b]a)[/b] Prove that $ \left( |z+1|-\sqrt 2 \right)\cdot \left( |z-1|-\sqrt 2 \right)\le 0,\quad\forall z\in C. $ [b]b)[/b] Prove that for any twelve numbers from $ C, $ namely $ z_1,\ldots ,z_{12} , $ there exist another twelve numbers $ \varepsilon_1,\ldots ,\varepsilon_{12}\in\{-1,1\} $ such that $$ \sum_{k=1}^{12} \left| z_k+\varepsilon_k \right| <17. $$

2015 Mathematical Talent Reward Programme, MCQ: P 14

Tags: algebra , complex number , coordinate geometry

$z=x+i y$ where $x$ and $y$ are two real numbers. Find the locus of the point $(x, y)$ in the plane, for which $\frac{z+i}{z-i}$ is purely imaginary (that is, it is of the form $i b$ where $b$ is a real number). [Here, $i=\sqrt{-1}$ [list=1] [*] A straight line [*] A circle [*] A parabole [*] None of these [/list]

2009 India IMO Training Camp, 5

Tags: algebra , complex number , polynomial

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

1973 Putnam, B2

Tags: complex number , rational number

Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.

2004 Postal Coaching, 12

Tags: algebra , complex number

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.

2012 AIME Problems, 13

Tags: vector , symmetry , trig identities , law of cosines , complex number , trigonometry

Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.

2009 Romania Team Selection Test, 3

Tags: trigonometry , floor function , algebra , complex analysis , complex number , function , inequalities

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

Gheorghe Țițeica 2025, P1

Tags: complex number

Find all complex numbers $a,b,c\in\mathbb{C}^*$ such that $$|a\overline{b}+b\overline{c}+c\overline{a}|=|a|^2+|b|^2+|c|^2.$$ [i]Mihai Opincariu[/i]

2008 IMS, 2

Tags: parallelogram , complex analysis , vector , complex number , geometry , function

Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

2013 BMT Spring, 4

Tags: algebra , complex number

Given a complex number $z$ satisfies $\operatorname{Im}(z)=z^2-z$, find all possible values of $|z|$.