Olympiad problems

2023 CIIM, 5

Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.

2005 Flanders Junior Olympiad, 3

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

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

2016 Miklós Schweitzer, 1

Tags: analytic number theory , number theory , complex number

For which complex numbers $\alpha$ does there exist a completely multiplicative, complex-valued arithmetic function $f$ such that \[ \sum_{n<x}f(n)=\alpha x+O(1)\,\,? \]

2010 Gheorghe Vranceanu, 2

Tags: complex number , complex geometry , geometry

Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $ [b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that $$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$ if $ \triangle $ is equilateral. [b]b)[/b] Show that $ \triangle $ is equilateral if $$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$

2007 Iran MO (3rd Round), 6

Tags: polynomial , quadratic , complex number , algebra

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. [img]http://i14.tinypic.com/4tk6mnr.png[/img] a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.

2016 District Olympiad, 2

Tags: absolute value , complex number , geometry , analytic geometry

Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying: $$ a^2+b^2+c^2-ab-bc-ca=0. $$ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.

2021 Science ON all problems, 1

Tags: algebra , complex number , inequality

Consider the complex numbers $x,y,z$ such that $|x|=|y|=|z|=1$. Define the number $$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$ $\textbf{(a)}$ Prove that $a$ is a real number. $\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$. [i] (Stefan Bălăucă & Vlad Robu)[/i]

2006 Petru Moroșan-Trident, 2

Tags: complex number , algebra

Consider $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that have the same nonzero modulus, and which verify $$ 0=\Re\left( \sum_{a=1}^n\sum_{b=1}^n\sum_{c=1}^n\sum_{d=1}^n \frac{z_bz_c}{z_az_d} \right) . $$ Prove that $ n\left( -1+\left| z_1 \right|^2 \right) =\sum_{k=1}^n\left| 1-z_k \right| . $ [i]Botea Viorel[/i]

2004 Germany Team Selection Test, 1

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

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

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 \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

2005 USAMTS Problems, 3

Tags: geometry , analytic geometry , quadratic , trapezoid , complex number , algebra , quadratic formula

Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.

2013 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , complex number

Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold: \begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*}

2011 ELMO Shortlist, 8

Tags: polynomial , induction , complex number , algebra

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]

2000 Vietnam National Olympiad, 3

Tags: polynomial , complex number , algebra

Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.

1996 USAMO, 1

Tags: trigonometry , ratio , complex number , geometric series , algebra

Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.

1994 National High School Mathematics League, 1

Tags: complex number

In the equation $x^2+z_1x+z_2+m=0$, $z_1,z_2,m$ are complex numbers, and $z_1^2-4z_2=16+20\text{i}$. Two roots of the equations are $\alpha,\beta$. If $|\alpha-\beta|=2\sqrt7$, find the maximum and minumum value of $|m|$.

1999 China Second Round Olympiad, 2

Tags: trigonometry , complex number , algebra

Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$.

1980 AMC 12/AHSME, 17

Tags: complex number

Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer? $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$

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________.

2014 NIMO Problems, 8

Tags: complex number

Let $a$, $b$, $c$, $d$ be complex numbers satisfying \begin{align*} 5 &= a+b+c+d \\ 125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\ 1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\ 25 &= a^4+b^4+c^4+d^4 \end{align*} Compute $abcd$. [i]Proposed by Evan Chen[/i]

2010 India IMO Training Camp, 10

Tags: geometry , parallelogram , inequalities , analytic geometry , trigonometry , complex number

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

2006 Mathematics for Its Sake, 3

Tags: complex number , complex inequality , algebra , induction , inequalities

Let be two complex numbers $ a,b $ chosen such that $ |a+b|\ge 2 $ and $ |a+b|\ge 1+|ab|. $ Prove that $$ \left| a^{n+1} +b^{n+1} \right|\ge \left| a^{n} +b^{n} \right| , $$ for any natural number $ n. $ [i]Alin Pop[/i]

2009 ELMO Problems, 5

Tags: trigonometry , complex number , geometry

Let $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of $\triangle ABD$. Prove that $\cos^2(\angle GOM)$ is rational and determine its value. [i]Evan o'Dorney[/i]

1999 National High School Mathematics League, 8

Tags: complex number

If $\theta=\arctan \frac{5}{12}$, $z=\frac{\cos 2\theta+\text{i}\sin2\theta}{239+\text{i}}$, then $\arg z=$________.

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]