Found problems: 563
2010 Harvard-MIT Mathematics Tournament, 4
Suppose that there exist nonzero complex numbers $a$, $b$, $c$, and $d$ such that $k$ is a root of both the equations $ax^3+bx^2+cx+d=0$ and $bx^3+cx^2+dx+a=0$. Find all possible values of $k$ (including complex values).
1995 National High School Mathematics League, 2
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
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 $QIP$.
2011 N.N. Mihăileanu Individual, 1
Let be a natural number $ n\ge 2, $ two complex numbers $ p,q, $ and four matrices $ A,B,C,D\in\mathcal{M}_n(\mathbb{C}) $ such that $ A+B=C+D=pI,AB+CD=qI $ and $ ABCD=0. $ Show that $ BCDA=0. $
[i]Marius Cavachi[/i]
2009 Purple Comet Problems, 16
Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$
2014 District Olympiad, 1
Solve for $z\in \mathbb{C}$ the equation :
\[ |z-|z+1||=|z+|z-1|| \]
2019 Jozsef Wildt International Math Competition, W. 45
Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties:
[list]
[*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$
[*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$
[/list]
Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
2019 PUMaC Team Round, 13
Let $e_1, e_2, . . . e_{2019}$ be independently chosen from the set $\{0, 1, . . . , 20\}$ uniformly at random.
Let $\omega = e^{\frac{2\pi}{i} 2019}$. Determine the expected value of $$|e_1\omega + e_2\omega^2 + ... + e_{2019}\omega^{2019}|.$$
2013 IMO Shortlist, G5
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
1980 VTRMC, 8
Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$
(a) Find two such complex numbers.
(b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$
(c) Show that $|z^2n -1|$ is rational for every $n.$
2005 Taiwan TST Round 1, 1
Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$
My solution was nearly complete...
1993 China National Olympiad, 4
We are given a set $S=\{z_1,z_2,\cdots ,z_{1993}\}$, where $z_1,z_2,\cdots ,z_{1993}$ are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide $S$ into some groups such that the following conditions are satisfied:
(1) Each element in $S$ belongs and only belongs to one group;
(2) For any group $p$, if we use $T(p)$ to denote the sum of all memebers in $p$, then for any memeber $z_i (1\le i \le 1993)$ of $p$, the angle between $z_i$ and $T(p)$ does not exceed $90^{\circ}$;
(3) For any two groups $p$ and $q$, the angle between $T(p)$ and $T(q)$ exceeds $90^{\circ}$ (use the notation introduced in (2)).
2005 SNSB Admission, 3
Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $
Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $
2016 District Olympiad, 3
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 $
2024 CMI B.Sc. Entrance Exam, 3
(a) FInd the number of complex roots of $Z^6 = Z + \bar{Z}$
(b) Find the number of complex solutions of $Z^n = Z + \bar{Z}$ for $n \in \mathbb{Z}^+$
2016 HMNT, 3
Complex number $\omega$ satisfies $\omega^5 = 2$. Find the sum of all possible values of $\omega^4 + \omega^3 + \omega^2 + \omega + 1$.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a,b,c,x,y,$ and $z$ be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If $xy+yz+xz=67$ and $x+y+z=2010$, find the value of $xyz$.
2002 VJIMC, Problem 1
Find all complex solutions to the system
\begin{align*}
(a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\
(a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\
(1+i)a+2ic&=0.\end{align*}
1994 APMO, 2
Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.
2019 PUMaC Team Round, 7
For all sets $A$ of complex numbers, let $P(A)$ be the product of the elements of $A$. Let $S_z = \{1, 2, 9, 99, 999, \frac{1}{z},\frac{1}{z^2}\}$, let $T_z$ be the set of nonempty subsets of $S_z$ (including $S_z$), and let $f(z) = 1 + \sum_{s\in T_z} P(s)$. Suppose $f(z) = 6125000$ for some complex number $z$. Compute the product of all possible values of $z$.
2012 Indonesia TST, 3
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.
2019 AIME Problems, 10
For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial
\[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \]
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of
\[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \]
can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2013 Kosovo National Mathematical Olympiad, 1
Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$
2004 All-Russian Olympiad, 2
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.