Olympiad problems

2014 NIMO Problems, 3

Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.) [i]Proposed by David Altizio[/i]

2013 Harvard-MIT Mathematics Tournament, 20

Tags: hmmt , algebra , polynomial , complex number

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

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*}

1993 AMC 12/AHSME, 20

Tags: quadratic , complex number , algebra , quadratic formula

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

1969 Czech and Slovak Olympiad III A, 4

Tags: complex number , algebra , absolute value , inequalities

Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.

1996 ITAMO, 5

Tags: complex number , geometry

Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.

2012 Bogdan Stan, 2

Tags: inequalities , algebra , complex number

Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $

2004 India IMO Training Camp, 1

Tags: parallelogram , complex number , circumcircle , geometry

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

1994 National High School Mathematics League, 2

Tags: complex number

Give two statements: (1) $a,b,c$ are complex numbers, if $a^2+b^2>c^2$, then $a^2+b^2-c^2>0$. (2) $a,b,c$ are complex numbers, if $a^2+b^2-c^2>0$, then $a^2+b^2>c^2$. Then, which is true? $\text{(A)}$ (1) is correct, (2) is correct as well $\text{(B)}$ (1) is correct, (2) is incorrect $\text{(C)}$ (1) is incorrect, (2) is incorrect as well $\text{(D)}$ (1) is incorrect, (2) is correct

2024 Mexican University Math Olympiad, 3

Tags: multiplicative function , greatest common divisor , euclidean distance , complex number , number theory

Consider a multiplicative function $ f $ from the positive integers to the unit disk centered at the origin, that is, $ f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} $ such that $ f(mn) = f(m)f(n) $. Prove that for every $ \epsilon > 0 $ and every integer $ k > 0 $, there exist $ k $ distinct positive integers $ a_1, a_2, \dots, a_k $ such that $ \text{gcd}(a_1, a_2, \dots, a_k) = k $ and $ d(f(a_i), f(a_j)) < \epsilon $ for all $ i, j = 1, \dots, k $.

1991 Arnold's Trivium, 66

Tags: complex analysis , complex number , logarithm

Solve the Dirichlet problem \[\Delta u=0\text{ for }x^2+y^2<1\] \[u=1\text{ for }x^2+y^2=1,\;y>0\] \[u=-1\text{ for }x^2+y^2=1,\;y<0\]

2021 BMT, T4

Tags: complex number , algebra

Let $z_1$, $z_2$, and $z_3$ be the complex roots of the equation $(2z -3\overline{z})^3 = 54i+54$. Compute the area of the triangle formed by $z_1$, $z_2$, and $z_3$ when plotted in the complex plane.

2016 HMNT, 3

Tags: algebra , complex number

Complex number $\omega$ satisfies $\omega^5 = 2$. Find the sum of all possible values of $\omega^4 + \omega^3 + \omega^2 + \omega + 1$.

2001 Stanford Mathematics Tournament, 5

Tags: college , quadratic , algebra , polynomial , complex number

What quadratic polynomial whose coeﬃcient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)

2013 Online Math Open Problems, 41

Tags: quadratic , trigonometry , function , complex number , geometry

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

1986 China National Olympiad, 3

Tags: complex number , algebra

Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.

2016 District Olympiad, 3

Tags: complex number , algebra

Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression $$ |\alpha x +\beta y| +|\alpha x-\beta y| $$ in each of the following cases: [b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $ [b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $

1947 Putnam, B4

Tags: complex number , polynomial

Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$

2012 Math Prize For Girls Problems, 13

Tags: complex number

For how many integers $n$ with $1 \le n \le 2012$ is the product \[ \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right) \] equal to zero?

2016 AIME Problems, 7

Tags: complex number

For integers $a$ and $b$ consider the complex number \[\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a, b)$ such that this complex number is a real number.

1992 National High School Mathematics League, 5

Tags: complex number , geometry

Points on complex plane that complex numbers $z_1,z_2$ corresponding to are $A,B$, and $|z_1|=4,4z_1^2-2z_1z_2+z_2^2=0$. $O$ is original point, then the area of $\triangle OAB$ is $\text{(A)}8\sqrt3\qquad\text{(B)}4\sqrt3\qquad\text{(C)}6\sqrt3\qquad\text{(D)}12\sqrt3$

2009 Romania National Olympiad, 2

Tags: minimal , complex number , inequalities , complex disc

Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $

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]

2008 Alexandru Myller, 2

Tags: complex number , trigonometry , binomial theorem , binomial coefficient , number theory

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

1996 National High School Mathematics League, 8

Tags: complex number

On the complex plane, non-zero complex numbers $z_1,z_2$ are on the circle with center $\text{i}$, radius of $1$. The real part of $\overline{z_1}\cdot z_2$ is $0$, and $\arg (z_1)=\frac{\pi}{6}$, then $z_2=$________.