Found problems: 563
What is the sum of the possible values for the complex number $a$ such that the coefficient of the $x^5$ term in the power series expansion of $\tfrac{x^3+ax^2+3x-4}{2x^2+ax+2}$ is $1?$
[b]2.[/b] Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. [b](F. 2)[/b]
A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule
\[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}}
\]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 2005\qquad
\textbf{(E)}\ 2^{2005}$
A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.
Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of
\begin{tabular}{|ccc|}
$b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\
$n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\
$n(b - a)$ & $n^2(a - b)$ & $a + n^3b$
\end{tabular}
Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$. Find $m+n$.
Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$.
b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.
Let $Z,W,\lambda$ be complex numbers, $|\lambda|\neq1$. Which statements are correct about the equation $\overline{Z}-\lambda Z=W$?
I. $Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}$ is a solution to the equation.
II. The equation has only one solution.
III. The equation has two solutions.
IV. The equation has infinitely many solutions.
$\text{(A)}$ Only I and II.
$\text{(B)}$ Only I and III.
$\text{(C)}$ Only I and IV.
$\text{(D)}$ None of $\text{(A)(B)(C)}$.
Answer the following questions:
(1) For complex numbers $\alpha ,\ \beta$, if $\alpha \beta =0$, then prove that $\alpha =0$ or $\beta =0$.
(2) For complex number $\alpha$, if $\alpha^2$ is a positive real number, then prove that $\alpha$ is a real number.
(3) For complex numbers $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}\ (n=1,\ 2,\ \cdots)$, assume that $\alpha_1\alpha_2,\ \cdots ,\ \alpha_k\alpha_{k+1},\ \cdots,\ \alpha_{2n}\alpha_{2n+1}$ and $\alpha_{2n+1}\alpha_1$ are all positive real numbers. Prove that $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}$ are all real numbers.
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that \[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]
Prove that whatever the complex number $z$ is, it is true that
$$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$
Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
Let be a complex number $ z $ having the property that $ \Re \left( z^n \right) >\Im \left( z^n \right) , $ for any natural numbers $ n. $
Show that $ z $ is a positive real number.
[i]Laurențiu Panaitopol[/i]
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$$ a^2+b^2+c^2-ab-bc-ca=0. $$
Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.
If $z_1$ , $z_2$ are the roots of the equation with real coefficients $z^2+az+b = 0$, prove that $ z^n_1 + z^n_2$ is a real number for any natural value of $n$. If particular of the equation $z^2 - 2z + 2 = 0$, express, as a function of $n$, the said sum.
Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that
$$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $
[i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]
Answer wither (i) or (ii):
(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$
(ii) Let $(a_{ij})$ be a matrix such that
$$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$
for all $i.$ Show that the determinant is not equal to $0.$
Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $\frac{x}{2\sin 36^\circ}$.
Consider $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that have the same nonzero modulus, and which verify
$$ 0=\Re\left( \sum_{a=1}^n\sum_{b=1}^n\sum_{c=1}^n\sum_{d=1}^n \frac{z_bz_c}{z_az_d} \right) . $$
Prove that $ n\left( -1+\left| z_1 \right|^2 \right) =\sum_{k=1}^n\left| 1-z_k \right| . $
[i]Botea Viorel[/i]
Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$ with at most $\frac{\ln(n)}{100}$ elements.
Define $f(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}$. Show that for some $r \ne 0$ we have $|f(r)| \geq \frac{|A|}{2}$.