Olympiad problems

2020 Kosovo National Mathematical Olympiad, 3

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

1983 Tournament Of Towns, (051) 3

Tags: circumcenter , projection , geometry , vector , incenter

The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are drawn rom $O$ on the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle ABC$. (V Galperin, Moscow)

1987 IMO Shortlist, 5

Tags: minimization , geometry , triangle , circumcenter , circumcircle

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

Mexican Quarantine Mathematical Olympiad, #4

Tags: geometry , altitude , circumcenter , orthocenter

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]

2024 Philippine Math Olympiad, P7

Tags: orthocenter , geometry , circumcenter , circumcircle

Let $ABC$ be an acute triangle with orthocenter $H$, circumcenter $O$, and circumcircle $\Omega$. Points $E$ and $F$ are the feet of the altitudes from $B$ to $AC$, and from $C$ to $AB$, respectively. Let line $AH$ intersect $\Omega$ again at $D$. The circumcircle of $DEF$ intersects $\Omega$ again at $X$, and $AX$ intersects $BC$ at $I$. The circumcircle of $IEF$ intersects $BC$ again at $G$. If $M$ is the midpoint of $BC$, prove that lines $MX$ and $OG$ intersect on $\Omega$.

2006 Sharygin Geometry Olympiad, 8.6

Tags: projection , circumcenter , interior , circumcircle , geometry

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.

2017 All-Russian Olympiad, 8

Tags: geometry , incircle , geometric transformation , incenter , circumcenter , reflection , circumcircle

In a non-isosceles triangle $ABC$,$O$ and $I$ are circumcenter and incenter,respectively.$B^\prime$ is reflection of $B$ with respect to $OI$ and lies inside the angle $ABI$.Prove that the tangents to circumcirle of $\triangle BB^\prime I$ at $B^\prime$,$I$ intersect on $AC$. (A. Kuznetsov)

2024 Brazil Team Selection Test, 2

Tags: midpoint , circumcenter , tangent , isosceles , geometry

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.

2009 Belarus Team Selection Test, 1

Tags: geometry , circumcenter , median , angle bisector

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

2012 Singapore Junior Math Olympiad, 3

Tags: collinear , geometry , 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.

Kyiv City MO Juniors Round2 2010+ geometry, 2017.9.1

Tags: symmetry , excenter , circumcenter , geometry

Find the angles of the triangle $ABC$, if we know that its center $O$ of the circumscribed circle and the center $I_A$ of the exscribed circle (tangent to $BC$) are symmetric wrt $BC$. (Bogdan Rublev)

2015 Saudi Arabia GMO TST, 3

Tags: geometry , orthocenter , circumcenter , perpendicular

Let $BD$ and $CE$ be altitudes of an arbitrary scalene triangle $ABC$ with orthocenter $H$ and circumcenter $O$. Let $M$ and $N$ be the midpoints of sides $AB$, respectively $AC$, and $P$ the intersection point of lines $MN$ and $DE$. Prove that lines $AP$ and $OH$ are perpendicular. Liana Topan

2003 Croatia Team Selection Test, 2

Tags: circumcircle , geometry , circumcenter

Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$

2016 Federal Competition For Advanced Students, P1, 2

Tags: geometry , orthocenter , circumcenter , circumcircle

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)

2007 Junior Tuymaada Olympiad, 4

Tags: circumcenter , angle , circumcircle , orthocenter , incenter , geometry

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

2004 India IMO Training Camp, 1

Tags: circumcircle , circumcenter , triangle , excenter , 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]

2007 Sharygin Geometry Olympiad, 18

Tags: orthocenter , locus , locus problems , circumcenter , circumcircle , geometry

Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

2006 Sharygin Geometry Olympiad, 11

Tags: geometry , circumcircle , concurrency , circumcenter

In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $ A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.

2020 Ukrainian Geometry Olympiad - December, 5

Tags: circumcircle , centroid , circumcenter , parallel , 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 $BC$.

2018-IMOC, G5

Tags: geometry , circumcenter , parallel , euler line , orthocenter , incenter

Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.

2013 Korea Junior Math Olympiad, 5

Tags: midpoint , perpendicular , acute , circumcenter , geometry , circumcircle

In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

2017 China Western Mathematical Olympiad, 3

Tags: circumcenter , geometry

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.

2004 Germany Team Selection Test, 2

Tags: geometry , circumcenter , triangle , excenter , circumcircle

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]

2021-IMOC qualification, G1

Tags: incenter , projection , circumcenter , circumcircle , geometry

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

2008 Oral Moscow Geometry Olympiad, 6

Tags: circumcenter , geometry , similar triangles , 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)