Olympiad problems

2002 IMC, 11

Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$. Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.

1991 Arnold's Trivium, 66

Tags: complex analysis , complex number , logarithm

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

1976 Miklós Schweitzer, 5

Tags: complex number , real analysis

Let $ S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) $, where the $ b_j$ are arbitrary and the $ z_j$ are nonzero complex numbers . Prove that \[ |S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.\] [i]G. Halasz[/i]

2022 CHMMC Winter (2022-23), 3

Tags: algebra , complex number , inequalities

Suppose that $a,b,c$ are complex numbers with $a+b+c = 0$, $|abc| = 1$, $|b| = |c|$, and $$\frac{9-\sqrt{33}}{48} \le \cos^2 \left( arg \left( \frac{b}{a} \right) \right)\le \frac{9+\sqrt{33}}{48} .$$ Find the maximum possible value of $|-a^6+b^6+c^6|$.

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 $1000a + 100b + 10c + d$.

1949 Putnam, A3

Tags: complex number , series

Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$, and that $|a_r - a_s| >1$ for $r\ne s.$ Show that the series $$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$ converges.

2006 Romania Team Selection Test, 1

Tags: circumcircle , geometric transformation , rotation , similar triangles , complex number , geometry

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

2013 NIMO Problems, 5

Tags: vieta , function , algebra , polynomial , complex number , system of equations

Let $x,y,z$ be complex numbers satisfying \begin{align*} z^2 + 5x &= 10z \\ y^2 + 5z &= 10y \\ x^2 + 5y &= 10x \end{align*} Find the sum of all possible values of $z$. [i]Proposed by Aaron Lin[/i]

1986 French Mathematical Olympiad, Problem 3

Tags: complex number , inequalities , complex inequality , algebra

(a) Prove or find a counter-example: For every two complex numbers $z,w$ the following inequality holds: $$|z|+|w|\le|z+w|+|z-w|.$$(b) Prove that for all $z_1,z_2,z_3,z_4\in\mathbb C$: $$\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.$$

2019 AMC 12/AHSME, 21

Tags: complex number

Let $$z=\frac{1+i}{\sqrt{2}}.$$ What is $$(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?$$ $\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$

1984 AMC 12/AHSME, 30

Tags: trigonometry , absolute value , complex number

For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then \[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \] equals $\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$

1983 Bulgaria National Olympiad, Problem 5

Tags: polynomial , solve equation , complex number , algebra

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

2008 iTest Tournament of Champions, 5

Tags: complex number

Let $c_1,c_2,c_3,\ldots, c_{2008}$ be complex numbers such that \[|c_1|=|c_2|=|c_3|=\cdots=|c_{2008}|=1492,\] and let $S(2008,t)$ be the sum of all products of these $2008$ complex numbers taken $t$ at a time. Let $Q$ be the maximum possible value of \[\left|\dfrac{S(2008,1492)}{S(2008,516)}\right|.\] Find the remainder when $Q$ is divided by $2008$.

1971 Spain Mathematical Olympiad, 5

Tags: complex number , limit , analysis

Prove that whatever the complex number $z$ is, it is true that $$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$ Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds $$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$

2012 Romanian Masters In 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]

2022 Nigerian Senior MO Round 2, Problem 3

Tags: geometry , complex number , imaginary triangle

In triangle $ABC$, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD, DE $ and $EC$ are $1, 3 $ and $5$ respectively. Find the length of $AC$.

2006 All-Russian Olympiad, 6

Tags: incenter , circumcircle , trapezoid , complex number , geometry

Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.

2022 China Team Selection Test, 5

Tags: inequalities , complex number

Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions: (1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$. (2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$. Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.

2025 China Team Selection Test, 18

Tags: complex number , inequalities

Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]

the 11th XMO, 10

Tags: complex number , algebra

Given $t\in\mathbb C$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate $$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.$$

1997 National High School Mathematics League, 15

Tags: complex number

$a_1,a_2,a_3,a_4,a_5$ are non-zero complex numbers, satisfying: $\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases}$ Where $S$ is a real number that $|S|\leq2$ Prove that points that $a_1,a_2,a_3,a_4,a_5$ refers to in the complex plane are concyclic.

1977 IMO, 1

Tags: complex number , geometry , square , polygon

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

2012 China Team Selection Test, 1

Tags: inequalities , complex number , triangle inequality , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

2009 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt , complex number , logarithm

Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

1989 IMO Longlists, 29

Tags: algebra , functional equation , complex number , function

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 \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]