The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.

On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.

The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that: $ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$

Let $\omega$ be the incircle of triangle $ABC$. For any positive real number $\lambda$, let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$. Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\angle C$ closer to $BC$. Similarly define $Y$ and $Z$. Let $\epsilon = \frac{1}{2024}$. Show that the circumcircle of triangle $XYZ$ lies inside $\omega_{1-\epsilon}$. [i]Note. Weaker results with smaller $\epsilon$ may be awarded points depending on the value of the constant $\epsilon <\frac{1}{2024}$.[/i]

In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?

Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Let $A$ and $B$ be two points on the circle with diameter $[CD]$ and on the different sides of the line $CD.$ A circle $\Gamma$ passing through $C$ and $D$ intersects $[AC]$ different from the endpoints at $E$ and intersects $BC$ at $F.$ The line tangent to $\Gamma$ at $E$ intersects $BC$ at $P$ and $Q$ is a point on the circumcircle of the triangle $CEP$ different from $E$ and satisfying $|QP|=|EP|. \: AB \cap EF =\{R\}$ and $S$ is the midpoint of $[EQ].$ Prove that $DR$ is parallel to $PS.$

Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$. a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$. b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be: $ \textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad \\ \textbf{(C)}\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad \\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad \\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.

Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively. a.Find $\triangle$ satisfy $S_{AMN}$ max b.$d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively. $d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$. [i]Dušan Djukić[/i]

Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$