Found problems: 1084
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.
The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.
Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?
$ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $
Answer the following questions :
[b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$
[b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$
[b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$
The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if
\[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]
The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$.
[i]Proposed by Luke Robitaille[/i]
Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$.
Proposed by Jeremy King, UK
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$.
[i]Alex Zhu.[/i]
Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let
$$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$
(1) If $n = 6063$, find the maximum value of $S$.
(2) If $n= 2021$, find the maximum value of $S$.
Let $k$ be a real number such that the system \begin{align*} &|25+20i-z|=5\\ &|z-4-k|=|z-3i-k| \\ \end{align*} has exactly one complex solution $z.$ The sum of all possible values of $k$ can be written as $\dfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Here $i=\sqrt{-1}.$
A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$.
Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.
Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.
Let $\triangle ABC$ be a triangle with circumcenter $O$. The perpendicular bisectors of the segments $OA,OB$ and $OC$ intersect the lines $BC,CA$ and $AB$ at $D,E$ and $F$, respectively. Prove that $D,E,F$ are collinear.
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.
Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that \[ XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.\]
Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.
Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is
$ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $
Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that:
$$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$
Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.
Given a triangle $ \triangle{ABC} $ with orthocenter $ H $. On its circumcenter, choose an arbitrary point $ P $ (other than $ A,B,C $) and let $ M $ be the mid-point of $ HP $. Now, we find three points $ D,E,F $ on the line $ BC, CA, AB $, respectively, such that $ AP \parallel HD, BP \parallel HE, CP \parallel HF $. Show that $ D, E, F, M $ are colinear.
Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is [i]patriotic[/i] if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is [i]dazzling[/i] if its endpoints are of different colors.
Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets.
[i]Ankan Bhattacharya[/i]
Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$