Olympiad problems

2009 All-Russian Olympiad Regional Round, 9.7

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.

2018 Yasinsky Geometry Olympiad, 3

Tags: geometry , construction , circumcenter , circumcircle , angle bisector , isosceles

Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$. (Andrey Mostovy)

2016 Thailand TSTST, 2

Tags: geometry , circumcenter , concurrency , tangent circle

Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.

2004 Estonia Team Selection Test, 2

Tags: geometry , parallel , equal segments , circumcenter

Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.

2016 Croatia Team Selection Test, Problem 3

Tags: circumcenter , midpoint , circumcircle , geometry

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 $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

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 $EF$.

2018 Czech-Polish-Slovak Junior Match, 4

Tags: collinear , geometry , circumcircle , circumcenter

A line passing through the center $M$ of the equilateral triangle $ABC$ intersects sides $BC$ and $CA$, respectively, in points $D$ and $E$. Circumcircles of triangle $AEM$ and $BDM$ intersects, besides point $M$, also at point $P$. Prove that the center of circumcircle of triangle $DEP$ lies on the perpendicular bisector of the segment $AB$.

2020 Ukrainian Geometry Olympiad - December, 5

Tags: parallel , centroid , circumcenter , circumcircle , geometry

Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

2012 Estonia Team Selection Test, 4

Tags: circumcircle , circumcenter , geometry , isosceles , equal segments

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

2021 239 Open Mathematical Olympiad, 2

Tags: geometry , concyclic , circumcenter , circumcircle

A triangle $ABC$ with an obtuse angle at the vertex $C$ is inscribed in a circle with a center at point $O$. Circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$, line $BC$ at points $B$ and $B_1$, and the perpendicular bisector of the segment $PC$ at points $D$ and $E$. Prove that points $D$ and $E$ together with the centers of the circumscribed circles of triangles $A_1OC$ and $B_1OC$ lie on one circle.

2020 China Northern MO, BP4

Tags: incenter , orthocenter , circumcenter , cyclic quadrilateral , spiral similarity , geometry

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.

1994 Spain Mathematical Olympiad, 2

Tags: geometry , circumcenter , sphere , 3-dimensional geometry , locus , 3d geometry

Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$. Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.

2004 Germany Team Selection Test, 2

Tags: excenter , circumcircle , circumcenter , triangle , geometry

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]

2016 Sharygin Geometry Olympiad, 7

Tags: construction , symmedian point , circumcenter , circumcircle , geometric transformation , reflection , geometry

Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point. [i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]

2019 Irish Math Olympiad, 5

Tags: geometry , reflection , concyclic , circumcenter

Let $M$ be a point on the side $BC$ of triangle $ABC$ and let $P$ and $Q$ denote the circumcentres of triangles $ABM$ and $ACM$ respectively. Let $L$ denote the point of intersection of the extended lines $BP$ and $CQ$ and let $K$ denote the reflection of $L$ through the line $PQ$. Prove that $M, P, Q$ and $K$ all lie on the same circle.

2024 Yasinsky Geometry Olympiad, 2

Tags: geometry , orthocenter , circumcenter , parallel

Let $ O $ and $ H $ be the circumcenter and orthocenter of the acute triangle $ ABC $. On sides $ AC $ and $ AB $, points $ D $ and $ E $ are chosen respectively such that segment $ DE $ passes through point $ O $ and $ DE \parallel BC $. On side $ BC $, points $ X $ and $ Y $ are chosen such that $ BX = OD $ and $ CY = OE $. Prove that $ \angle XHY + 2\angle BAC = 180^\circ $. [i]Proposed by Matthew Kurskyi[/i]

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

2017 JBMO Shortlist, G2

Tags: geometry , circumcenter , equal angles , circumcircle

Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $ M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.

2017 China Western Mathematical Olympiad, 3

Tags: geometry , circumcenter

D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.

2016 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: midpoint , geometry , perpendicular , circumcenter , orthocenter

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

2004 Germany Team Selection Test, 2

Tags: excenter , circumcircle , circumcenter , triangle , geometry

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]

2008 Oral Moscow Geometry Olympiad, 4

Tags: geometry , orthocenter , circumcenter , distance

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

2021-IMOC, G10

Tags: geometry , circumcenter , incenter , tangent circle

Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.

1992 All Soviet Union Mathematical Olympiad, 559

Tags: geometry , square , circumcenter

$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square.

2024 Brazil Team Selection Test, 2

Tags: geometry , circumcenter , midpoint , tangent , isosceles

Let $ ABC $ be an acute-angled scalene triangle with circumcenter $ O $. Denote by $ M $, $ N $, and $ P $ the midpoints of sides $ BC $, $ CA $, and $ AB $, respectively. Let $ \omega $ be the circle passing through $ A $ and tangent to $ OM $ at $ O $. The circle $ \omega $ intersects $ AB $ and $ AC $ at points $ E $ and $ F $, respectively (where $ E $ and $ F $ are distinct from $ A $). Let $ I $ be the midpoint of segment $ EF $, and let $ K $ be the intersection of lines $ EF $ and $ NP $. Prove that $ AO = 2IK $ and that triangle $ IMO $ is isosceles.