Found problems: 1084
Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$
Determine positive integers $n\geq 3$ such that there exists a set $M$ of $n$ complex numbers and a positive integer $m$ such that $(1+z_1z_2z_3)^m=1$ for all pairwise distinct $z_1,z_2,z_3\in M$.
[i]Vlad Matei[/i]
Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if
$$(u+ix)(v+iy)(w+iz)=i?$$
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $
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 be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $
Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $
Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly.
If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.
Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.
Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.
The triangle $ABC$ is given. Let $A'$ be the midpoint of the side $BC$, $B_c{}$ be the projection of $B{}$ onto the bisector of the angle $ACB{}$ and $C_b$ be the projection of the point $C{}$ onto the bisector of the angle $ABC$. Let $A_0$ be the center of the circle passing through $A', B_c, C_b$. The points $B_0$ and $C_0$ are defined similarly. Prove that the incenter of the triangle $ABC$ coincides with the orthocenter of the triangle $A_0B_0C_0$.
Let $P$ be a point in the plane of triangle $ABC$. Denote the reflections of $A,B,C$ over $P$ by $A',B'$ and $C'$, respectively. Let $A'',B'',C''$ be the reflection of $A',B',C'$ over $BC,CA$ and $AB$, respectively. Let the line $A''B''$ intersects $AC$ at $A_b$ and let $A''C''$ intersects $AB$ at $A_c$. Denote by $\omega_A$ the circle through the points $A,A_b,A_c$. The circles $\omega_B,\omega_C$ are defined similarly. Prove that $\omega_A ,\omega_B ,\omega_C$ are coaxial, i.e., they share a common radical axis.
[i]Proposed by Navneel Singhal, Delhi and K. V. Sudharshan, Chennai, India[/i]
How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$)
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.
(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$.
(2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation
\[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \]
has at least one complex root with norm at most $3n|\omega|$.
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
Isosceles triangle $ABC$ with $AB = AC$ is inscibed is a unit circle $\Omega$ with center $O$. Point $D$ is the reflection of $C$ across $AB$. Given that $DO = \sqrt{3}$, find the area of triangle $ABC$.
Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.
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 $CM$.
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$.
Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds.
$$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$
[i]Costică Ambrinoc[/i]
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 $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$
Let $a\in (0,1)$, $f(z)=z^2-z+a, z\in \mathbb{C}$. Prove the following statement holds:
For any complex number z with $|z| \geq 1$, there exists a complex number $z_0$ with $|z_0|=1$, such that $|f(z_0)| \leq |f(z)|$.