Olympiad problems

1985 Spain Mathematical Olympiad, 5

Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.

1993 National High School Mathematics League, 7

Tags: complex number , quadratic , imaginary numbers , algebra

Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.

2014 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , quadratic , complex number , dilation , geometric transformation , geometry , inequalities

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

2006 ITAMO, 3

Tags: complex number , geometry , reflection , parallelogram , geometric transformation

Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.

ICMC 3, 2

Tags: complex number , sum , roots of unity

Find integers $a$ and $b$ such that \[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\] [i]proposed by the ICMC Problem Committee[/i]

2014 China Team Selection Test, 5

Tags: complex number , inequalities

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

2025 District Olympiad, P4

Tags: complex number , geometry

Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$. [list=a] [*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$. [*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.

2020 ISI Entrance Examination, 1

Tags: complex number , polynomial , 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 $f(i+\omega)=0$ .

1986 IMO Longlists, 29

Tags: rotation , complex number , geometry , vector , geometric transformation

We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M \star O) = (O \star I)\star M$ holds?

2014 Taiwan TST Round 2, 2

Tags: complex number , geometry , 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.

2013 Harvard-MIT Mathematics Tournament, 25

Tags: hmmt , complex number

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

2007 ISI B.Math Entrance Exam, 3

Tags: complex number , limit , polynomial , algebra

For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$.

2012 District Olympiad, 3

Tags: complex number , sequence , trigonometry , algebra

Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and $$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$ for all natural $ k\ge 2. $ [b]a)[/b] Find $ a_2. $ [b]b)[/b] Determine this sequence.

2013 District Olympiad, 3

Tags: abstract algebra , complex number , linear algebra , algebra , matrix

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

2011 IMO Shortlist, 2

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

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]

2008 iTest Tournament of Champions, 5

Tags: complex number

Let $c_1,c_2,c_3,\ldots, c_{2008}$ be complex numbers such that \[|c_1|=|c_2|=|c_3|=\cdots=|c_{2008}|=1492,\] and let $S(2008,t)$ be the sum of all products of these $2008$ complex numbers taken $t$ at a time. Let $Q$ be the maximum possible value of \[\left|\dfrac{S(2008,1492)}{S(2008,516)}\right|.\] Find the remainder when $Q$ is divided by $2008$.

2013 IMO Shortlist, G5

Tags: complex number , hexagon , geometry

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.

1959 Miklós Schweitzer, 7

Tags: complex number , real analysis

[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series $\sum_{n=1}^{\infty} \epsilon_n z_n$ is convergente. [b](F. 9)[/b]

2025 ISI Entrance UGB, 4

Tags: mapping , composition , complex number , periodicity , inductive definition , algebra

Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the [i]period[/i] of $z$. Determine the total number of points in $S^1$ of period $2025$. (Hint : $2025 = 3^4 \times 5^2$)

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]

2006 Victor Vâlcovici, 2

Tags: algebra , complex number , complex number geometry

Prove that the affixes of three pairwise distinct complex numbers $ z_0,z_1,z_2 $ represent an isosceles triangle with right angle at $ z_0 $ if and only if $ \left( z_1-z_0 \right)^2 =-\left( z_2-z_0 \right)^2. $

2004 Gheorghe Vranceanu, 4

Tags: complex number , linear algebra , matrix

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]

2012 International Zhautykov Olympiad, 2

Tags: complex number , geometry

Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.

2017 CHMMC (Fall), 4

Tags: complex number

Let $a = e^{\frac{4\pi i}5}$ be a nonreal fifth root of unity and $b = e^{\frac{2\pi i}{17}}$ be a nonreal seventeenth root of unity. Compute the value of the product \[(a + b) (a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]

2014 Harvard-MIT Mathematics Tournament, 31

Tags: complex number , ratio , geometric series , trigonometry

Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]