Found problems: 11
2005 Gheorghe Vranceanu, 2
Prove that the sum of the $ \text{2005-th} $ powers of three pairwise distinct complex numbers is the imaginary unit if their modulus are equal and the sum of these numbers is the imaginary unit.
2019 Romanian Master of Mathematics, 2
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
2016 Mathematical Talent Reward Programme, MCQ: P 5
$ABCD$ is a quadrilateral on complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
[list=1]
[*] Rectangle
[*] Rhombus
[*] Isosceles Trapezium
[*] Square
[/list]
2007 Gheorghe Vranceanu, 2
Let be a natural number $ n\ge 2 $ and an imaginary number $ z $ having the property that $ |z-1|=|z+1|\cdot\sqrt[n]{2} . $ Denote with $ A,B,C $ the points in the Euclidean plane whose representation in the complex plane are the affixes of $
z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} , $ respectively. Prove that $ AB $ is perpendicular to $ AC. $
2019 Romania National Olympiad, 3
Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation
$$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$
represent the vertices of an equilateral triangle in the complex plane.
2003 Gheorghe Vranceanu, 3
Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $
1985 Spain Mathematical Olympiad, 5
Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.
2019 Romanian Masters In Mathematics, 2
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
2003 Romania National Olympiad, 3
Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $
[i]Gheorghe Iurea[/i]
2016 Romania National Olympiad, 3
[b]a)[/b] Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $
[b]b)[/b] Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $
2010 Gheorghe Vranceanu, 2
Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $
[b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that
$$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$
if $ \triangle $ is equilateral.
[b]b)[/b] Show that $ \triangle $ is equilateral if
$$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$