Olympiad problems

2016 Saudi Arabia BMO TST, 1

Given two non-constant polynomials $P(x),Q(x)$ with real coefficients. For a real number $a$, we define $$P_a= \{z \in C : P(z) = a\}, Q_a =\{z \in C : Q(z) = a\}$$ Denote by $K$ the set of real numbers $a$ such that $P_a = Q_a$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.

1959 Putnam, A3

Tags: function , complex , functional equation

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$.

2019 LIMIT Category A, Problem 9

Tags: geometry , complex number , complex , algebra

$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a $\textbf{(A)}~\text{Rectangle}$ $\textbf{(B)}~\text{Rhombus}$ $\textbf{(C)}~\text{Isosceles Trapezium}$ $\textbf{(D)}~\text{Square}$

2007 Thailand Mathematical Olympiad, 10

Tags: algebra , complex

Find the smallest positive integer $n$ such that the equation $\sqrt3 z^{n+1} - z^n - 1 = 0$ has a root on the unit circle.

2018 China Girls Math Olympiad, 5

Tags: algebra , maximum value , complex

Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.

2015 QEDMO 14th, 6

Tags: algebra , complex , complex number

Let $n\ge 2$ be an integer. Let $z_1, z_2,..., z_n$ be complex numbers in such a way that for all integers $k$ with $1\le k\le n$: $$\Pi_{i = 1,i\ne k}^{n} (z_k- z_i) = \Pi_{i = 1,i\ne k}^{n} (z_k+ z_i).$$ Show that two of them are the same.

2008 Postal Coaching, 3

Tags: inequalities , algebra , complex

Let $a$ and $b$ be two complex numbers. Prove the inequality $$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$

2013 ELMO Shortlist, 7

Tags: ratio , trigonometry , complex , circumcircle , collinear , parallelogram , geometry

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

1993 Romania Team Selection Test, 1

Tags: algebra , sequence , complex

Consider the sequence $z_n = (1+i)(2+i)...(n+i)$. Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.

2013 ELMO Shortlist, 7

Tags: geometry , ratio , circumcircle , parallelogram , trigonometry , collinear , complex

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

1949-56 Chisinau City MO, 57

Tags: algebra , complex

Solve the equation: $| z |- 2 = 1 + 2 i$, where $| r |$ is the modulus of a complex number $z$.