Found problems: 1084
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.
Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\][i]Vasile Pop[/i]
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$
Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $
[b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that
$$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$
for any nonnegative integer $ n. $
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$
for any complex number $x.$
Let $z$ be a complex number such that $z,z^2,z^3$ are all collinear in the complex plane . Show that $z$ is a real number .
[list=a]
[*]Let $A$ be a matrix from $\mathcal{M}_{2}(\mathbb{C})$, $A\neq aI_{2}$,
for any $a\in\mathbb{C}$. Prove that the matrix $X$ from $\mathcal{M}
_{2}(\mathbb{C})$ commutes with $A$, that is, $AX=XA$, if and only if there
exist two complex numbers $\alpha$ and $\alpha^{\prime}$, such that $X=\alpha
A+\alpha^{\prime}I_{2}$.
[*]Let $A$, $B$ and $C$ be matrices from $\mathcal{M}_{2}(\mathbb{C})$, such
that $AB\neq BA$, $AC=CA$ and $BC=CB$. Prove that $C$ commutes with all
matrices from $\mathcal{M}_{2}(\mathbb{C})$.[/list]
Regular $2n$-gon is inscribed in the unit circle. Find the sum of the squares of all sides and diagonal lengths in the $2n$-gon.
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$
$\textbf{(A) } 20 \qquad
\textbf{(B) } 21 \qquad
\textbf{(C) } 22 \qquad
\textbf{(D) } 23 \qquad
\textbf{(E) } 24 $
Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent.
[i]Proposed by Romania, Radu-Andrew Lecoiu[/i]
Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.
Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality:
$$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$
Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that
\[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]
There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that
\[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]
[b]a)[/b] Show that the solution of the equation $ |z-i|=1 $ in $ \mathbb{C} $ is the set $ \{ 2e^{i\alpha} \sin\alpha |\alpha\in [0,\pi ) \} . $
[b]b)[/b] Let be $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that verify the inequalities
$$ \left| z_k-i \right|\le 1,\quad\forall k\in\{ 1,2,\ldots ,n \} . $$
Prove that there exists a complex number $ w $ such that $ |w-i|\le 1 $ and $ w^n=z_1z_2\cdots z_n. $
[i]Dan-Ștefan Marinescu[/i]
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that
\[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}
\]
Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear.
[i]Proposed by Holden Mui[/i]