Found problems: 1084
The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.
Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.
Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.
Find all polynomials $f(x)$ with real coefficients for which
\[f(x)f(2x^2) = f(2x^3 + x).\]
Let $n\ge 2$ be a natural number. Find all complex numbers $z$ which simultaneously satisfy the relations
$\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;$
$\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.$
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position?
$ \textbf{(A)}\ \frac {1}{6} \qquad \textbf{(B)}\ \frac {1}{5} \qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$
There are $2n$ complex numbers that satisfy both $z^{28}-z^{8}-1=0$ and $|z|=1$. These numbers have the form $z_{m}=\cos\theta_{m}+i\sin\theta_{m}$, where $0\leq\theta_{1}<\theta_{2}< \dots <\theta_{2n}<360$ and angles are measured in degrees. Find the value of $\theta_{2}+\theta_{4}+\dots+\theta_{2n}$.
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]
Let $m,n\geq2$ be positive integers, and let $a_1,a_2,\ldots ,a_n$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_1,e_2,\ldots,e_n$, not all zero, with $\left|{\,e}_i\,\right|<m$ for all $i$, such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$.
Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1
a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional
b)Does the conclusion from the previous exercise remains true for $n=4$?
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
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]
Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$.
[list=a]
[*] Show that lines $PE$ and $QD$ meet on $\omega$.
[*] Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well.
[/list]
[i]Merlijn Staps[/i]
Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$.
[i]Author: Charles Leytem, Luxembourg[/i]
$\alpha,\beta$ are conjugate complex numbers. If $|\alpha-\beta|=2\sqrt3$, $\frac{\alpha}{\beta^2}$ is a real number, then $|\alpha|=$________.
Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.
Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$.
[i]By Ali Khezeli[/i]
Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$.
[i]T. F. Mori, G. J. Szekely[/i]
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?
Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$