Found problems: 3882
2013 IMAC Arhimede, 5
Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$.
2019 Singapore Senior Math Olympiad, 1
In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$
1985 All Soviet Union Mathematical Olympiad, 408
The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points. Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius.
2013 NIMO Problems, 8
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Michael Ren[/i]
2006 Romania Team Selection Test, 1
Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral.
[i]Valentin Vornicu[/i]
2025 Japan MO Finals, 2
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABO$ and $ACO$, respectively. The circumcircle of $\triangle AO_1O_2$ intersects segment $BC$ at two distinct points $P$ and $Q$, with the four points $B, P, Q, C$ appearing in this order along $BC$. Let $O_3$ be the circumcenter of $\triangle OPQ$. Prove that points $A, O, O_3$ are collinear.
2018 Taiwan TST Round 3, 2
Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$.
Show that $FG$ is perpendicular to $IO$.
1994 IMO, 2
Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.
2012 Estonia Team Selection Test, 3
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.
Brazil L2 Finals (OBM) - geometry, 2009.2
Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.
2018 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.
2005 AIME Problems, 14
In triangle $ABC$, $AB=13$, $BC=15$, and $CA=14$. Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$
2004 Nicolae Coculescu, 4
Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides.
[b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $
[b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $
[i]Geoghe Duță[/i]
2004 Iran Team Selection Test, 3
Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.
2000 Poland - Second Round, 2
Bisector of angle $BAC$ of triangle $ABC$ intersects circumcircle of this triangle in point $D \neq A$. Points $K$ and $L$ are orthogonal projections on line $AD$ of points $B$ and $C$, respectively. Prove that $AD \ge BK + CL$.
1998 IMO Shortlist, 1
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
2009 All-Russian Olympiad, 2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
2020 Regional Olympiad of Mexico Southeast, 5
Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.
2011 IMO Shortlist, 5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.
[i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]
Kharkiv City MO Seniors - geometry, 2017.11.5
The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.
2022 Bosnia and Herzegovina IMO TST, 1
Let $ABC$ be a triangle such that $AB=AC$ and $\angle BAC$ is obtuse. Point $O$ is the circumcenter of triangle $ABC$, and $M$ is the reflection of $A$ in $BC$. Let $D$ be an arbitrary point on line $BC$, such that $B$ is in between $D$ and $C$. Line $DM$ cuts the circumcircle of $ABC$ in $E,F$. Circumcircles of triangles $ADE$ and $ADF$ cut $BC$ in $P,Q$ respectively. Prove that $DA$ is tangent to the circumcircle of triangle $OPQ$.
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality
\[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\]
and determine when equality holds.
2019 Korea Winter Program Practice Test, 2
$\omega_1,\omega_2$ are orthogonal circles, and their intersections are $P,P'$. Another circle $\omega_3$ meets $\omega_1$ at $Q,Q'$, and $\omega_2$ at $R,R'$. (The points $Q,R,Q',R'$ are in clockwise order.) Suppose $P'R$ and $PR'$ meet at $S$, and let $T$ be the circumcenter of $\triangle PQR$. Prove that $T,Q,S$ are collinear if and only if $O_1,S,O_3$ are collinear. ($O_i$ is the center of $\omega_i$ for $i=1,2,3$.)
2002 Iran MO (2nd round), 4
Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$
2010 Sharygin Geometry Olympiad, 5
A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.