Found problems: 563
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $
[i]Ion Tecu[/i]
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.
[i]Proposed by Michael Kural[/i]
[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
[b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:
\[3\le |z-a|+|z-b|+|z-c|\le 4,\]
for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]
Let $a,b,c$ be real numbers, $z_{1},z_{2},z_{3}$ be complex numbers such that
$\begin{cases}
\displaystyle|z_1|=|z_2|=|z_3|=1\\
\displaystyle\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1\\
\end{cases}$
Find $|az_{1}+bz_{2}+cz_{3}|$.
Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?
Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.
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. $
Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $
Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center]
Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$
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.
In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.
For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying
$(x+iy)^n+(x-iy)^n = 2x^n$ .
(a) Determine $S(n)$ for $n = 2,3,4$.
(b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
$z$ is a complex number that $\left|2z+\frac{1}{z}\right|=1$, then the range value of $\arg(z)$ is________.
Solve for $z\in \mathbb{C}$ the equation :
\[ |z-|z+1||=|z+|z-1|| \]
Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying
\[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]
Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?
A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients
Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression
$$ |\alpha x +\beta y| +|\alpha x-\beta y| $$
in each of the following cases:
[b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $
[b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $
Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$