Olympiad problems

2010 Contests, 2

Tags: number theory , real analysis , quadratic , complex number , sum of roots , algebra , polynomial

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2012 Romanian Masters In Mathematics, 2

Tags: trigonometry , geometry , complex number , circumcircle

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]

2010 Harvard-MIT Mathematics Tournament, 5

Tags: complex number

Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^2+y^2$.

2022 CHMMC Winter (2022-23), 3

Tags: inequalities , complex number , algebra

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 $|-a^6+b^6+c^6|$.

2021 China Second Round Olympiad, Problem 3

Tags: conic , ellipse , complex number

There exists complex numbers $z=x+yi$ such that the point $(x, y)$ lies on the ellipse with equation $\frac{x^2}9+\frac{y^2}{16}=1$. If $\frac{z-1-i}{z-i}$ is real, compute $z$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)[/i]

1977 IMO, 1

Tags: geometry , complex number , polygon , square

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.

2006 Mathematics for Its Sake, 3

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

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]

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 $z_2=$________.

2014 NIMO Problems, 3

Tags: modular arithmetic , algebra , polynomial , complex number

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]

2018 AIME Problems, 6

Tags: complex number

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

1995 National High School Mathematics League, 2

Tags: complex number

Complex numbers of apexes of 20-regular polygon inscribed to unit circle refer to are $Z_1,Z_2,\cdots,Z_{20}$ on complex plane. Then the number of points in $Z_1^{1995},Z_2^{1995},\cdots,Z_{20}^{1995}$ refer to is $\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}10\qquad\text{(D)}20$

2024 Pan-American Girls’ Mathematical Olympiad, 6

Tags: geo , complex number , barycentric coordinates , geometry

Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$. The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$. Prove that $U$ is the centroid of triangle $QIP$.

2004 All-Russian Olympiad, 2

Tags: incenter , exterior angle , complex number , geometry

Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.

2018 Purple Comet Problems, 14

Tags: complex number

A complex number $z$ whose real and imaginary parts are integers satisfies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$ .

2007 China National Olympiad, 1

Tags: complex number , inequalities

Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that \[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]

1990 National High School Mathematics League, 5

Tags: complex number

Two non-zero-complex numbers $x,y$, satisfy that $x^2+xy+y^2=0$. Then the value of $(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}$ is $\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}$none above

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]

2009 India IMO Training Camp, 5

Tags: complex number , polynomial , algebra

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

2008 Harvard-MIT Mathematics Tournament, 4

Tags: complex number , trigonometry

Suppose that $ a, b, c, d$ are real numbers satisfying $ a \geq b \geq c \geq d \geq 0$, $ a^2 \plus{} d^2 \equal{} 1$, $ b^2 \plus{} c^2 \equal{} 1$, and $ ac \plus{} bd \equal{} 1/3$. Find the value of $ ab \minus{} cd$.

1956 Miklós Schweitzer, 2

Tags: complex number

[b]2.[/b] Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. [b](F. 2)[/b]

2007 Purple Comet Problems, 10

Tags: trigonometry , complex number

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

2016 Mathematical Talent Reward Programme, MCQ: P 3

Tags: complex number

$z$ is a complex number and $|z|=1$ and $z^2\neq 1$. Then $\frac{z}{1-z^2}$ lies on [list=1] [*] A line not passing through origin [*] $|z|=2$ [*] $x$-axis [*] $y$-axis [/list]

2011 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt , complex number , trigonometry

Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

2018 PUMaC Live Round, Misc. 2

Tags: complex number

What is the sum of the possible values for the complex number $a$ such that the coefficient of the $x^5$ term in the power series expansion of $\tfrac{x^3+ax^2+3x-4}{2x^2+ax+2}$ is $1?$

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?