Found problems: 1084
The point $M$ is the middle of the side $BC$ of the acute-angled triangle $ABC$ and the points $E$ and $F$ are respectively perpendicular foot of $M$ to the sides $AC$ and $AB$. The points $X$ and $Y$ lie on the plane such that $\triangle XEC\sim\triangle CEY$ and $\triangle BYF\sim\triangle XBF$(The vertices of triangles with this order are corresponded in the similarities) and the points $E$ and $F$ [u]don't[/u][neither] lie on the line $XY$. Prove that $XY\perp AM$.
Let $ABC$ be a triangle with circumcentre $O$ and centroid $G$. Let $M$ be the midpoint of $BC$ and $N$ the reflection of $M$ across $O$. Prove that $NO = NA$ if and only if $\angle AOG = 90^{\circ}$.
[i]Proposed by Pranjal Srivastava[/i]
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.
[i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]
Circle touches parallelogram‘s $ABCD$ borders $AB, BC$ and $CD$ respectively at points $K, L$ and $M$. Perpendicular is drawn from vertex $C$ to $AB$ . Prove, that the line $KL$ divides this perpendicular into two equal parts (with the same length).
Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $OH$ is parallel to $BC$. Let $AH$ intersects again with the circumcircle of $ABC$ at $X$, and let $XB, XC$ intersect with $OH$ at $Y, Z$, respectively. If the projections of $Y,Z$ to $AB,AC$ are $P,Q$, respectively, show that $PQ$ bisects $BC$.
[i]Proposed by usjl[/i]
Prove that
$$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$
for any natural number $ n. $
[i]Mihai Piticari[/i]
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ lying in the interior. Let $E$ and $F$ be the midpoints of the segments $BC$ and $AD$, respectively. Let $X$ be the point lying on the same side of the line $EF$ as the vertex $C$ such that $\triangle EXF$ and $\triangle BOA$ are similar. Prove that $XC = XD$.
$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule:
[list]
[*]It always moves clockwise.
[*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.)[/list]
Given that it reaches all leaves at least once. Show $n$ cannot be odd.
[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series
$\sum_{n=1}^{\infty} \epsilon_n z_n$
is convergente. [b](F. 9)[/b]
Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$
satisfy $|z_3|=|z_4|=1$
Let $ (x,y)$ be a pair of real numbers satisfying \[ 56x \plus{} 33y \equal{} \frac{\minus{}y}{x^2\plus{}y^2}, \qquad \text{and} \qquad 33x\minus{}56y \equal{} \frac{x}{x^2\plus{}y^2}.
\]Determine the value of $ |x| \plus{} |y|$.
Set $M=\{z\in\mathbb{C}|(z-1)^2=|z-1|^2\}$, then
$\text{(A)}M=\{\text{pure imaginary number}\}$
$\text{(B)}M=\{\text{real number}\}$
$\text{(C)}M=\{\text{real number}\}\subset M\subset\{\text{complex number}\}$
$\text{(D)}M=\{\text{complex number}\}$
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #14[/i]
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate
and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.
Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
[/hide]
Let be two complex numbers $ a,b. $ Show that the following affirmations are equivalent:
$ \text{(i)} $ there are four numbers $ x_1,x_2,x_3,x_4\in\mathbb{C} $ such that $ \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, $ and
$$ x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a,\quad\forall j_1\in\{ 1,2\} ,\quad\forall j_2\in\{ 3,4\} . $$
$ \text{(ii)} a^3=b^3 $ or $ b=\overline{a} $ (the conjugate of a).
Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy
\begin{align*}
\vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}}
\end{align*}
$(b)$ Find all real numbers $x$ that satisfy following equation $:$
\begin{align*}
\frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6}
\end{align*}
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?
Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$
If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that
[b]a)[/b] Triangle $A_1A_3A_5$ is equilateral
[b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]
In a triangle $ABC$, let $O$ be the circumcenter, $H$ the orthocenter, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.