Found problems: 1084
In $\triangle ABC$,$AB=AC>BC$, point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively,$G $ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$, then $H$ is the incenter of $\triangle GBC$.
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$.
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$.
[i]Evan O'Dorney.[/i]
$T_1$ consists of a single branch from 0 to 1 in the complex plane. This branch then splits into two branches at the endpoint which each form a $135^\circ$ angle with the branch in $T_1$, and each branch has length $\frac{1}{\sqrt{2}}$. This process is repeated so that from each terminal branch in $T_n$, we form two more branches at angles $135^\circ$ with $\frac{1}{\sqrt{2}}$ the length. Let $L_n$ be the collection of the $2^{n-1}$ endpoints of the tree $T_n$. If multiple terminal branches end at the same point, then that point is counted multiple times in $L_n$. Shown below is $T_k$ for $k=1, 2, 3$ with dots at the points in $L_k$. Find the sum of $\ell^2$ over all points $\ell$ in $L_{10}$.
Let $a, b, c, x, y$ and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}$. If $xy+yz+zx=1000$ and $x+y+z=2016$, find the value of $xyz$.
Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic.
[i]Carl Lian.[/i]
Complex number $z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)$. Points that three complex numbers $z,(1+\text{i})z,2\overline{z}$ refer to on complex plane are $P,Q,R$. When $P,Q,R$ are not collinear, $PQSR$ is a parallelogram. The longest distance between $S$ and the original point is________.
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
[asy]
import olympiad;
size(120);
defaultpen(linewidth(0.8));
pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A;
pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y);
draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4"));
draw(rightanglemark(A,B,C,3));
draw(A--B--C--cycle);
draw((0,2)--(0,0)--(2,0),Arrows(size=8));
label("$6$",(PAx+PBx)/2,S);
label("$8$",(PAy+PCy)/2,W);
[/asy]
Let $a$ and $b$ be complex numbers satisfying the two equations
\begin{align*}
a^3 - 3ab^2 & = 36 \\
b^3 - 3ba^2 & = 28i.
\end{align*}
Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a| = M$.
Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.
Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .
Find one complex value of $x$ that satisfies the equation $\sqrt{3}x^7+x^4+2=0$.
On the complex plane, non-zero complex numbers $z_1,z_2$ are on the circle with center $\text{i}$, radius of $1$. The real part of $\overline{z_1}\cdot z_2$ is $0$, and $\arg (z_1)=\frac{\pi}{6}$, then $z_2=$________.
A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have
$$
|f(x)+g(y)| = | f(y) + g(x)|.
$$
Proposed by [i]Mojtaba Zare, Amirabbas Mohammadi[/i]
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
Let $a$,$b$,$c$ be real numbers.
Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$
Find $|az_{1}+bz_{2}+cz_{3}|$.
Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\overline{BD}$ such that $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there exists a point $Y$ on segment $\overline{AC}$ such that $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$.
For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial
\[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \]
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of
\[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \]
can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that
\[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2}
\]
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.