Olympiad problems

2015 Kyiv Math Festival, P4

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

1988 Tournament Of Towns, (169) 2

Tags: symmetry , circumcenter , geometry

We are given triangle $ABC$. Two lines, symmetric with $AC$, relative to lines $AB$ and $BC$ are drawn, and meet at $K$ . Prove that the line $BK$ passes through the centre of the circumscribed circle of triangle $ABC$. (V.Y. Protasov)

2020-IMOC, G2

Tags: circumcenter , concurrency , geometry

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)

2001 IMO Shortlist, 2

Tags: geometric inequality , circumcenter , triangle , geometry

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

2019 Romanian Master of Mathematics Shortlist, G3

Tags: orthocenter , incenter , circumcenter , geometry

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

2018 Yasinsky Geometry Olympiad, 3

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

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)

2017 JBMO Shortlist, G2

Tags: circumcenter , equal angles , circumcircle , geometry

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 $MNK$.

2016 Regional Olympiad of Mexico Center Zone, 4

Tags: circles , parallelogram , circumcenter , geometry

Let $A$ be one of the two points where the circles whose centers are the points $M$ and $N$ intersect. The tangents in $A$ to such circles intersect them again in $B$ and $C$, respectively. Let $P$ be a point such that the quadrilateral $AMPN$ is a parallelogram. Show that $P$ is the circumcenter of triangle $ABC$.

2019 Romanian Master of Mathematics Shortlist, G1

Tags: circumcenter , median , circumcircle , geometry

Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$. Aleksandr Kuznetsov, Russia

2019 Oral Moscow Geometry Olympiad, 3

Tags: concyclic , circumcenter , circumcircle , geometry , orthocenter

In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and $H$ is the orthocenter. The angle bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

Champions Tournament Seniors - geometry, 2012.2

Tags: equal angles , geometry , circumcenter , collinear

About the triangle $ABC$ it is known that $AM$ is its median, and $\angle AMC = \angle BAC$. On the ray $AM$ lies the point $K$ such that $\angle ACK = \angle BAC$. Prove that the centers of the circumcircles of the triangles $ABC, ABM$ and $KCM$ lie on the same line.

2019 Bulgaria National Olympiad, 2

Tags: circumcenter , orthocenter , geometry , perpendicular bisector , circumcircle

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$

1987 IMO Longlists, 22

Tags: geometry , triangle , minimization , 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]

Kyiv City MO Seniors Round2 2010+ geometry, 2019.11.3

Tags: circumcenter , midline , geometry

The line $\ell$ is perpendicular to the side $AC$ of the acute triangle $ABC$ and intersects this side at point $K$, and the circumcribed circle $\vartriangle ABC$ at points $P$ and $T$ (point P on the other side of line $AC$, as the vertex $B$). Denote by $P_1$ and $T_1$ - the projections of the points $P$ and $T$ on line $AB$, with the vertices $A, B$ belong to the segment $P_1T_1$. Prove that the center of the circumscribed circle of the $\vartriangle P_1KT_1$ lies on a line containing the midline $\vartriangle ABC$, which is parallel to the side $AC$. (Anton Trygub)

2017 Finnish National High School Mathematics Comp, 5

Tags: circumcenter , diameter , circles , 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 $T$.

2001 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: symmetry , equal segments , circumcenter , perpendicular , geometry

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.

2024 Yasinsky Geometry Olympiad, 2

Tags: geometry , parallel , circumcenter , orthocenter

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]

2003 IMO Shortlist, 3

Tags: circumcenter , triangle , excenter , geometry , 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]

2023 Yasinsky Geometry Olympiad, 5

Tags: geometry , circumcenter , construction

Let $ABC$ be a scalene triangle. Given the center $I$ of the inscribe circle and the points $K_1$, $K_2$ and $K_3$ where the inscribed circle is tangent to the sides $BC$, $AC$ and $AB$. Using only a ruler, construct the center of the circumscribed circle of triangle $ABC$. (Hryhorii Filippovskyi)

2009 Tournament Of Towns, 5

Tags: geometry , rhombus , circumcenter

In rhombus $ABCD$, angle $A$ equals $120^o$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ so that angle $NAM$ equals $30^o$. Prove that the circumcenter of triangle $NAM$ lies on a diagonal of of the rhombus.

2019 Pan-African, 3

Tags: circumcircle , geometry , circumcenter , equilateral triangle

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

2019 Tuymaada Olympiad, 2

Tags: midpoint , geometry , trapezoid , circumcenter , distance

A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.

2014 Sharygin Geometry Olympiad, 13

Tags: geometry , locus , circumcenter

Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.

Geometry Mathley 2011-12, 16.4

Tags: incenter , circumcenter , orthocenter , geometry

A triangle $ABC$ is inscribed in the circle $(O)$, and has incircle $(I)$. The circles with diameter $IA$ meets $(O)$ at $A_1$ distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Line $B_1C_1$ meets $BC$ at $A_2$, and points $B_2,C_2$ are defined in the same manner. Prove that $O$ is the orthocenter of triangle $A_2B_2C_2$. Trần Minh Ngọc

2016 Sharygin Geometry Olympiad, 7

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

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]