Found problems: 1084
For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.
The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.
Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines.
With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$.
[i]Proposed by Vilmos Molnár-Szabó, Budapest[/i]
For real numbers $x_0,x_1,\cdots,x_n$, there exists real numbers $y_0,y_1,\cdots,y_n$, satisfying that $z_0^2=z_1^2+z_2^2+\cdots+z_n^2$, where $z_k=x_k+\text{i}y_{k}(k=0,1,\cdots,n)$. Find all such $(x_0,x_1,\cdots,x_n)$.
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
Let $x,y,z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha =\beta =\gamma$.
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$.
Remark: Remember that the roots of a polynomial are not necessarily real numbers.
Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
Let $\omega$ be a primitive $7$th root of unity. Find
$$\prod_{k=0}^6\left(1+\omega^k-\omega^{2k}\right).$$
(A complex number is a primitive root of unity if and only if it can be written in the form $e^{2k\pi i/n}$, where $k$ is relatively prime to $n$.)
Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and
\[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\]
If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.
Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and
\[
AB \cdot DE = BC \cdot EF = CD \cdot FA.
\]
Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.
Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.
Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point.
[i]Proposed by Aaron Thomas, UK[/i]
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 $x^2+y^2$.
Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.
Determine a necessary and sufficient condition for the affixes of three complex numbers $z_1$ , $z_2$ and $z_3$ are the vertices of an equilateral triangle.
Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $
Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that
[list]
[*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and
[*] $\angle BEN = \angle MED$.
[/list]
[i]Proposed by Evan Chen[/i]
What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$
In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following:
[list]
[*] For marked points $X,Y$, mark the reflection of $X$ across $Y$.
[*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$.
[*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$.
[/list]
No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves,
a) The circumcenter of $\triangle ABC$.
b) The incenter of $\triangle ABC$.
Proposed by [i]Deniz Can Karaçelebi[/i]
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
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 $\arg x_1 - \arg x_2 = 98^{\circ}.$
Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.