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 $\zeta$.

2024 IMC, 1

Tags: complex number

Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$.

MathLinks Contest 1st, 2

Tags: complex number , inequalities , algebra

Let $m$ be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements $1$, there exists a subset having the modulus of the sum of the elements in the subset greater than $m$. Prove that $$\frac14 \le m \le \frac12.$$ (Optional Task for 3p) Find a smaller value for the RHS.

1998 IMO Shortlist, 6

Tags: trigonometry , analytic geometry , geometry , complex number

Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

2009 District Olympiad, 4

Tags: complex number , absolute value , geometry

[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle. [b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.

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.

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]

2020 China Team Selection Test, 1

Tags: complex number , algebra , roots of unity

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

1999 Polish MO Finals, 3

Tags: trigonometry , analytic geometry , geometry , complex number

Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

2002 Moldova Team Selection Test, 4

Tags: complex number , polynomial , irreducible , divisibility , algebra

Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).

2019 Romania National Olympiad, 3

Tags: complex number , geometry , complex geometry , algebra

Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation $$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$ represent the vertices of an equilateral triangle in the complex plane.

2004 Gheorghe Vranceanu, 4

Tags: complex number , algebra , equation

Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $

2005 AIME Problems, 9

Tags: trigonometry , floor function , modular arithmetic , complex number

For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$?

1986 IMO Longlists, 29

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

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?

2023 District Olympiad, P2

Tags: geometry , complex number

Let $ABC$ be an equilateral triangle. On the small arc $AB{}$ of its circumcircle $\Omega$, consider the point $N{}$ such that the small arc $NB$ measures $30^\circ{}$. The perpendiculars from $N{}$ onto $AC$ and $AB$ intersect $\Omega$ again at $P{}$ and $Q{}$ respectively. Let $H_1,H_2$ and $H_3$ be the orthocenters of the triangles $NAB, QBC$ and $CAP$ respectively. [list=a] [*]Prove that the triangle $NPQ$ is equilateral. [*]Prove that the triangle $H_1H_2H_3$ is equilateral. [/list]

2018 Purple Comet Problems, 14

Tags: complex number

A complex number $z$ whose real and imaginary parts are integers satisfies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$ .

1967 IMO Shortlist, 3

Tags: trigonometry , algebra , product , calculate , complex number

Without using tables, find the exact value of the product: \[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]

2003 Alexandru Myller, 2

Tags: number theory , complex number , algebra , newton s binom

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

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 $n$.

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.

2012 Romanian Master of 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]

2015 Postal Coaching, Problem 4

Tags: geometry , complex number

Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of $$\frac{KM + LN}{AC + BD}$$ .

2013 Online Math Open Problems, 48

Tags: algebra , polynomial , calculus , integration , complex number

$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].) [i]Victor Wang[/i]

1983 IMO Longlists, 53

Tags: trigonometry , complex number , algebra

Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation \[\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}\] Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$

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?