Olympiad problems

2014 China Team Selection Test, 5

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

2015 NIMO Problems, 5

Tags: algebra , complex number , polynomial

Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials \[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \quad\text{ and }\quad Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] Let $S$ be the set of all complex numbers which are a root of [i]either[/i] $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$ [i]Proposed by Michael Tang[/i]

2012 Indonesia TST, 3

Tags: angle chasing , complex number , cyclic quadrilateral , geometry

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

2021 Science ON grade X, 3

Tags: complex number , newton sums , vieta , algebra

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

2019 LIMIT Category B, Problem 2

Tags: complex number , algebra

Let $\mathbb C$ denote the set of all complex numbers. Define $$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$ $\textbf{(B)}~A\subset B\text{ and }A\ne B$ $\textbf{(C)}~B\subset A\text{ and }B\ne A$ $\textbf{(D)}~\text{None of the above}$

2024 Harvard-MIT Mathematics Tournament, 8

Tags: complex number , algebra

Let $\zeta = \cos \frac {2pi}{13} + i \sin \frac {2pi}{13}$ . Suppose $a > b > c > d$ are positive integers satisfying $$|\zeta^a + \zeta^b + \zeta^c +\zeta^d| =\sqrt3.$$ Compute the smallest possible value of $1000a + 100b + 10c + d$.

2004 Germany Team Selection Test, 1

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

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

1983 AIME Problems, 2

Tags: complex number , absolute value , search , function

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

2012 Online Math Open Problems, 49

Tags: polynomial , induction , derivative , complex number , algebra , calculus

Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$. [i]Author: Alex Zhu[/i]

2000 Romania National Olympiad, 3

Tags: function , complex number , cartesian plane , geometry , analytic geometry , algebra

A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon. Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.

1992 AMC 12/AHSME, 28

Tags: quadratic formula , quadratic , complex number , algebra

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

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 $1/6$.

2004 France Team Selection Test, 3

Tags: analytic geometry , pigeonhole principle , complex number , triangle inequality , inequalities , geometry

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

2023 Iran MO (3rd Round), 1

Tags: complex number , geometry

Given $12$ complex numbers $z_1,...,z_{12}$ st for each $1 \leq i \leq 12$: $$|z_i|=2 , |z_i - z_{i+1}| \geq 1$$ prove that : $$\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}$$

1984 AMC 12/AHSME, 10

Tags: complex number

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i,-2+i$ and $-1-2i$. The fourth number is A. $2+i$ B. $2-i$ C. $1-2i$ D. $-1+2i$ E. $-2-i$

2013 Iran MO (3rd Round), 3

Tags: complex number , algebra

For every positive integer $n \geq 2$, Prove that there is no $n-$tuple of distinct complex numbers $(x_1,x_2,\dots,x_n)$ such that for each $1 \leq k \leq n$ following equality holds. $\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) $ (20 points)

2021 Romania National Olympiad, 1

Tags: complex number , algebra

Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.

2013 Math Prize For Girls Problems, 6

Tags: arithmetic sequence , geometric sequence , complex number , imaginary numbers , ratio

Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

1966 IMO Shortlist, 36

Tags: cyclic quadrilateral , complex number , geometry

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

1983 Bulgaria National Olympiad, Problem 5

Tags: solve equation , complex number , polynomial , algebra

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

1987 IMO Longlists, 59

Tags: trigonometry , complex number , algebra

It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$: \[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\] [b](a) [/b]Give one condition to make the system consistent. [b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$

2021 Science ON all problems, 3

Tags: complex number , newton sums , vieta , algebra

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

1997 IMC, 3

Tags: logarithm , complex number , trigonometry , absolute value , function , integration , floor function

Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.

2008 VJIMC, Problem 1

Tags: complex number , algebra , polynomial

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

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}$