Olympiad problems

2015 Balkan MO Shortlist, G5

Tags: circumcircle , geometry

Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.( Proposed by Emil Stoyanov ) [img]http://estoyanov.net/wp-content/uploads/2015/09/est.png[/img]

2013 ELMO Shortlist, 5

Tags: circumcircle , asymptote , geometry

Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic. [i]Proposed by Eric Chen[/i]

2000 Bulgaria National Olympiad, 2

Tags: circumcircle , geometry

Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.

1996 IMO Shortlist, 3

Tags: geometry , circumcircle , symmetry , orthocenter , trigonometry

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

2019 Bundeswettbewerb Mathematik, 3

Tags: circumcircle , square , geometry

Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively. Show that the lines $EF$ and $GH$ are parallel.

1963 Kurschak Competition, 3

Tags: circumcircle , geometric inequalities , geometry

A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

2010 Contests, 3

Tags: reflection , geometric transformation , geometry , circumcircle

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

2020 Italy National Olympiad, #4

Tags: circumcircle , geometry

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

1984 All Soviet Union Mathematical Olympiad, 381

Tags: geometry , congruent triangles , circumcircle

Given triangle $ABC$ . From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the circumscribed circle at $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane.

1992 Polish MO Finals, 1

Tags: geometry , circumcircle

Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.

2008 China Team Selection Test, 1

Tags: ratio , trigonometry , circumcircle , projective geometry , geometry

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

2010 Ukraine Team Selection Test, 11

Tags: geometry , geometric inequality , excircle , circumcircle

Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.

2017 Iran Team Selection Test, 5

Tags: circumcircle , geometry

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

2007 Danube Mathematical Competition, 2

Tags: parallelogram , circumcircle , geometry

Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.

2001 Macedonia National Olympiad, 3

Tags: trigonometry , geometry , symmetry , analytic geometry , projective geometry , circumcircle

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.

2014 ITAMO, 2

Tags: circumcircle , geometry , symmetry

Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that (a) $M$ is the midpoint of $AB$; (b) $N$ is the midpoint of $AC$; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$. Prove that $\angle APM = \angle PBA$.

2015 Ukraine Team Selection Test, 1

Tags: geometry , angle bisector , circumcircle , equal angles

Let $O$ be the circumcenter of the triangle $ABC, A'$ be a point symmetric of $A$ wrt line $BC, X$ is an arbitrary point on the ray $AA'$ ($X \ne A$). Angle bisector of angle $BAC$ intersects the circumcircle of triangle $ABC$ at point $D$ ($D \ne A$). Let $M$ be the midpoint of the segment $DX$. A line passing through point $O$ parallel to $AD$, intersects $DX$ at point $N$. Prove that angles $BAM$ and $CAN$ angles are equal.

2002 Tournament Of Towns, 4

Tags: geometry , radical axis , power of a point , circumcircle

Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.

2004 Romania Team Selection Test, 14

Tags: concurrency , circumcircle , geometry , ratio

Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

2015 Thailand TSTST, 2

Tags: circumcircle , parallel , hexagon , geometry

Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.

1995 Taiwan National Olympiad, 5

Tags: geometry , circumcircle

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

2009 Stars Of Mathematics, 2

Tags: circumcircle , geometry , circumcenter

Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

2003 Junior Balkan MO, 3

Tags: geometry , circumcircle , angle bisector

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

1992 IMO Longlists, 59

Tags: geometric inequality , circumcircle , polygon , geometry

Let a regular $7$-gon $A_0A_1A_2A_3A_4A_5A_6$ be inscribed in a circle. Prove that for any two points $P, Q$ on the arc $A_0A_6$ the following equality holds: \[\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .\]

2006 China Western Mathematical Olympiad, 2

Tags: circumcircle , geometry

$AB$ is a diameter of the circle $O$, the point $C$ lies on the line $AB$ produced. A line passing though $C$ intersects with the circle $O$ at the point $D$ and $E$. $OF$ is a diameter of circumcircle $O_{1}$ of $\triangle BOD$. Join $CF$ and produce, cutting the circle $O_{1}$ at $G$. Prove that points $O,A,E,G$ are concyclic.