Olympiad problems

1999 Romania National Olympiad, 3

Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$ satisfy $|z_3|=|z_4|=1$

1953 Czech and Slovak Olympiad III A, 1

Tags: complex number , plane

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.

2002 China Western Mathematical Olympiad, 3

Tags: geometry , perimeter , polynomial , complex number , algebra

In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.

2012 Online Math Open Problems, 7

Tags: trigonometry , complex number

A board $64$ inches long and $4$ inches high is inclined so that the long side of the board makes a $30$ degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a+b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. What is $a+b+c$? [i]Author: Ray Li[/i] [hide="Clarification"]The problem is intended to be a two-dimensional problem. The board's dimensions are 64 by 4. The long side of the board makes a 30 degree angle with the ground. One corner of the board is touching the ground.[/hide]

2012 Belarus Team Selection Test, 2

Tags: circumcircle , polynomial , quadratic , complex number , geometry , algebra

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

2004 District Olympiad, 4

Tags: complex number , algebra

If $x,y \in (0, \frac{\pi}{2})$ such as $ (cosx+isiny)^n=cos(nx)+isin(ny)$ for two consecutive positive integers, then the relation is true for all positive integers.

2015 AMC 12/AHSME, 24

Tags: probability , trigonometry , function , complex number

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number? $\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$

2017 District Olympiad, 4

Tags: complex number , algebra

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 $$ \sum_{k=1}^{12} \left| z_k+\varepsilon_k \right| <17. $$

2005 India Regional Mathematical Olympiad, 1

Tags: geometry , rhombus , parallelogram , complex number

Let ABCD be a convex quadrilateral; P,Q, R,S are the midpoints of AB, BC, CD, DA respectively such that triangles AQR, CSP are equilateral. Prove that ABCD is a rhombus. Find its angles.

2012 Kyrgyzstan National Olympiad, 3

Tags: complex number , geometry

Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?

2009 Romania Team Selection Test, 3

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

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.

2020 ISI Entrance Examination, 1

Tags: polynomial , complex number , algebra

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

1987 IMO Longlists, 59

Tags: trigonometry , complex number , algebra

It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$: \[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\] [b](a) [/b]Give one condition to make the system consistent. [b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$

2005 QEDMO 1st, 8 (Z2)

Tags: geometry , 3d geometry , rotation , complex number

Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.

1991 Vietnam Team Selection Test, 3

Tags: limit , linear algebra , matrix , vector , invariant , complex number , logarithm

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

2023 Iran MO (3rd Round), 1

Tags: complex number , geometry

Given $12$ complex numbers $z_1,...,z_{12}$ st for each $1 \leq i \leq 12$: $$|z_i|=2 , |z_i - z_{i+1}| \geq 1$$ prove that : $$\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}$$

2012 Math Prize For Girls Problems, 13

Tags: complex number

For how many integers $n$ with $1 \le n \le 2012$ is the product \[ \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right) \] equal to zero?

2024 Iran MO (3rd Round), 1

Tags: algebra , complex number , fe , function , functional equation

Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

2009 AIME Problems, 2

Tags: analytic geometry , rotation , complex number , geometry , similar triangles

There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that \[ \frac {z}{z \plus{} n} \equal{} 4i. \]Find $ n$.

1991 National High School Mathematics League, 11

Tags: complex number

For two complex numbers $z_1,z_2$ satisfy that $|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3$, then $\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=$________.

2010 Hong kong National Olympiad, 1

Tags: complex number , geometry

Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

2008 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: geometry , parallelogram , analytic geometry , vector , complex number

Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that: (i) Lines $ BC$ and $ AA'$ are orthogonal. (ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid

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]

2013 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt , complex number

Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.

2014 NIMO Problems, 5

Tags: circumcircle , ratio , trigonometry , complex number , pythagorean theorem , geometry

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]