Olympiad problems

2018 Romania National Olympiad, 2

Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$

2015 District Olympiad, 3

Tags: algebra , complex number , equation

Solve in $ \mathbb{C} $ the following equation: $ |z|+|z-5i|=|z-2i|+|z-3i|. $

1985 Balkan MO, 1

Tags: geometry , analytic geometry , graphing lines , complex number , circumcircle

In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.

2019 Romania National Olympiad, 4

Tags: polynomial , complex number , root , superior algebra

Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that $\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct. $\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$

2014 Singapore Senior Math Olympiad, 33

Tags: trigonometry , complex number

Find the value of $2(\sin2^{\circ}\tan1^{\circ}+\sin4^{\circ}\tan1^{\circ}+\cdots+\sin178^{\circ}\tan 1^{\circ})$

2005 Today's Calculation Of Integral, 11

Tags: integration , domain , complex number , function , algebra , logarithm , calculus

Calculate the following indefinite integrals. [1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$ [2] $\int \frac{e^x}{e^x+e^{a-x}}dx$ [3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$ [4] $\int x\ln (x^2-1)dx$ [5] $\int \frac{2(x+2)}{x^2+4x+1}dx$

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 $|\alpha|=$________.

1997 Romania National Olympiad, 4

Tags: complex number , polynomial , algebra

Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center] Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$

2020 Jozsef Wildt International Math Competition, W22

Tags: calculus , integration , complex analysis , complex number

Prove that $$\operatorname{Re}\left(\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)+\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)=\frac{7\pi^2}{72}-\frac{\ln^23}8$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ [i]Proposed by Paolo Perfetti[/i]

2004 Nicolae Coculescu, 3

Tags: algebra , complex number

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]

1981 Romania Team Selection Tests, 4.

Tags: complex number , algebra

Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$. Show that for every $a\in\mathbb{C}$ and $r>0$, the number \[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\] is an integer. Interpet this result geometrically. [i]Octavian Stănășilă[/i]

1993 National High School Mathematics League, 9

Tags: complex number

If $z\in\mathbb{C},\arg{(z^2-4)}=\frac{5}{6}\pi,\arg{(z^2+4)}=\frac{\pi}{3}$, then the value of $z$ is________.

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 $t$?

2001 Romania Team Selection Test, 1

Tags: algebra , complex number , trigonometry

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

2013 IMC, 5

Tags: complex number , function

Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime? [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

2015 Postal Coaching, Problem 4

Tags: complex number , geometry

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 $$\frac{KM + LN}{AC + BD}$$ .

2007 Grigore Moisil Intercounty, 4

Tags: function , group theory , complex number

Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that: $ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $ $ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $ $ \text{(3)} $ contain both of the above functions.

1991 Arnold's Trivium, 84

Tags: vector , quadratic , complex number , induction

Find the number of positive and negative squares in the canonical form of the quadratic form $\sum_{i<j}(x_i-x_j)^2$ in $n$ variables. The same for the form $\sum_{i<j}x_i x_j$.

2020 Purple Comet Problems, 12

Tags: complex number

There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.

2004 Germany Team Selection Test, 1

Tags: algebra , system of equations , polynomial , linear algebra , complex number , matrix

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

1985 ITAMO, 3

Tags: complex number

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

1967 IMO Shortlist, 3

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

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

1989 IMO Longlists, 29

Tags: algebra , functional equation , function , complex number

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

2014 Ukraine Team Selection Test, 3

Tags: geometry , complex number , 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.

2006 Petru Moroșan-Trident, 1

Tags: matrix , linear algebra , complex number

Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that $$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$ Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle. [i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]