Olympiad problems

2021 BMT, 14

Let $r_1, r_2, ..., r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute $$\sum^{47}_{i=1}r^{2020}_i .$$

2013 AIME Problems, 12

Tags: algebra , polynomial , trigonometry , vieta , complex number

Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

2005 Today's Calculation Of Integral, 11

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

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] $\int \frac{2(x+2)}{x^2+4x+1}dx$

2008 VJIMC, Problem 1

Tags: complex number , algebra , polynomial

Find all complex roots (with multiplicities) of the polynomial $$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$

1977 Miklós Schweitzer, 7

Tags: function , group theory , galois theory , complex number , complex analysis

Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous. [i]L. Szekelyhidi[/i]

2005 AMC 12/AHSME, 22

Tags: complex number

A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule \[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}} \]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 2005\qquad \textbf{(E)}\ 2^{2005}$

2019 Ramnicean Hope, 2

Tags: complex number , algebra

Let be three complex numbers $ a,b,c $ such that $ |a|=|b|=|c|=1=a^2+b^2+c^2. $ Calculate $ \left| a^{2019} +b^{2019} +c^{2019} \right| . $ [i]Costică Ambrinoc[/i]

1989 IMO Longlists, 70

Tags: trigonometry , complex number , algebra

Given that \[ \frac{\cos(x) \plus{} \cos(y) \plus{} \cos(z)}{\cos(x\plus{}y\plus{}z)} \equal{} \frac{\sin(x)\plus{} \sin(y) \plus{} \sin(z)}{\sin(x \plus{} y \plus{} z)} \equal{} a,\] show that \[ \cos(y\plus{}z) \plus{} \cos(z\plus{}x) \plus{} \cos(x\plus{}y) \equal{} a.\]

2005 Flanders Math Olympiad, 3

Tags: geometry , complex number , prime number , number theory

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1

Tags: algebra , polynomial , galois theory , complex number , superior algebra

Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

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.

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.

2003 Romania National Olympiad, 4

Tags: complex number , group theory , algebra , cyclic group

[b]a)[/b] Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number. [b]b)[/b] Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer. [i]Paltin Ionescu[/i]