Olympiad problems

Mathley 2014-15, 4

Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$. Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University

2007 District Olympiad, 1

Tags: geometry , equal segments , angles , Angle Chasing , Circumcenter , circumcircle

Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.

Estonia Open Senior - geometry, 2012.1.3

Tags: geometry , right triangle , median , Circumcenter

Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$. a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle. b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

2016 Middle European Mathematical Olympiad, 3

Tags: geometry , geometry proposed , Circumcenter , perpendicular lines

Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$. Prove that $\angle QCB = \angle PCO$.

1996 Argentina National Olympiad, 3

Tags: geometry , circumcircle , hexagon , Circumcenter , orthocenter

The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.

2011 Tournament of Towns, 1

Tags: geometry , Circumcenter , incenter , equal segments

$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.

2022 Korea National Olympiad, 2

Tags: geometry , circumcircle , Circumcenter , parallel

In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry , orthocenter , Circumcenter , circumcircle

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

2015 India Regional MathematicaI Olympiad, 1

Tags: geometry , Circumcenter , incenter , reflection

Let $ABC$ be a triangle. Let $B'$ denote the reflection of $b$ in the internal angle bisector $l$ of $\angle A$.Show that the circumcentre of the triangle $CB'I$ lies on the line $l$ where $I$ is the incentre of $ABC$.

2004 India IMO Training Camp, 1

Tags: geometry , circumcircle , Circumcenter , Triangle , IMO Shortlist , excenters , geometry solved

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

2004 Oral Moscow Geometry Olympiad, 4

Tags: geometry , Centroid , Circumcenter , equal segments

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

Mexican Quarantine Mathematical Olympiad, #4

Tags: geometry , altitude , orthocenter , Circumcenter

Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes of triangle $ABC$ opposite to vertices $A$, $B$, and $C$ respectively. Let $B_2$ and $C_2$ be the midpoints of $BB_1$ and $CC_1$, respectively. Let $O$ be the intersection of lines $BC_2$ and $CB_2$. Prove that $O$ is the circumcenter of triangle $ABC$ if and only if $H$ is the midpoint of $AA_1$. [i]Proposed by Dorlir Ahmeti[/i]

2012 Singapore Junior Math Olympiad, 3

Tags: geometry , collinear , excenter , Circumcenter

In $\vartriangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of $\vartriangle ABD$ and the points $C, D$ lie on the same straight line.

2017 Finnish National High School Mathematics Comp, 5

Tags: circle , Circumcenter , diameter , geometry

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

2019 Romanian Master of Mathematics Shortlist, G3

Tags: incenter , geometry , Circumcenter , orthocenter

Let $ABC$ be an acute-angled triangle with $AB \ne AC$, and let $I$ and $O$ be its incenter and circumcenter, respectively. Let the incircle touch $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Assume that the line through $I$ parallel to $EF$, the line through $D$ parallel to$ AO$, and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of the triangle $ABC$. Petar Nizic-Nikolac, Croatia

2016 Croatia Team Selection Test, Problem 3

Tags: geometry , geometry proposed , Circumcenter , midpoint , circumcircle

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

2004 Tournament Of Towns, 4

Tags: geometry , Locus , Circumcenter , circumcircle

A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.

Brazil L2 Finals (OBM) - geometry, 2010.2

Tags: geometry , parallelogram , circumcircle , Circumcenter

Let $ABCD$ be a parallelogram and $\omega$ be the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$ with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on $\omega$.

2020 China Northern MO, BP4

Tags: geometry , incenter , orthocenter , geometry solved , Circumcenter , cyclic quadrilateral , Spiral Similarity

In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.

2012 Balkan MO Shortlist, G4

Tags: geometry , incenter , circumcircle , Cyclic , Circumcenter , circles

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$. The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$, and $F$ is the midpoint of $TN$. Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$, respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$, and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$, respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.

2008 Oral Moscow Geometry Olympiad, 6

Tags: geometry , similar triangles , Circumcenter , collinear

Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$. (A. Zaslavsky)

2009 Belarus Team Selection Test, 1

Tags: geometry , Circumcenter , angle bisector , median

In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$. Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$ I. Voronovich

2013 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , perpendicular bisector , midpoint , Circumcenter

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.

2020-IMOC, G2

Tags: geometry , concurrent , concurrency , Circumcenter

Let $O$ be the circumcenter of triangle $ABC$. Define $O_{A0} = O_{B0} = O_{C0} = O$. Recursively, define $O_{An}$ to be the circumcenter of $\vartriangle BO_{A(n-1)}C$. Similarly define $O_{Bn}, O_{Cn}$. Find all $n \ge 1$ so that for any triangle $ABC$ such that $O_{An}, O_{Bn}, O_{Cn}$ all exist, it is true that $AO_{An}, BO_{Bn}, CO_{Cn}$ are concurrent. (Li4)

2025 Israel TST, P2

Tags: geometry , tangent , Circumcenter

Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$. Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$. Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$. Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$. Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$. Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.