Olympiad problems

2024 Israel National Olympiad (Gillis), P4

Tags: geometry , concurrence , Triangle , Circumcenter , geometric transformation , reflection

Acute triangle $ABC$ is inscribed in a circle with center $O$. The reflections of $O$ across the three altitudes of the triangle are called $U$, $V$, $W$: $U$ over the altitude from $A$, $V$ over the altitude from $B$, and $W$ over the altitude from $C$. Let $\ell_A$ be a line through $A$ parallel to $VW$, and define $\ell_B$, $\ell_C$ similarly. Prove that the three lines $\ell_A$, $\ell_B$, $\ell_C$ are concurrent.

2015 NZMOC Camp Selection Problems, 7

Tags: geometry , circumcircle , cyclic quadrilateral , Circumcenter

Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.

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.

2005 Sharygin Geometry Olympiad, 18

Tags: concurrency , concurrent , symmetry , lines , Circumcenter , geometry

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?

Croatia MO (HMO) - geometry, 2018.3

Tags: geometry , tangent , Circumcenter , Reflections , circumcircle

Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$

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]

2019 Argentina National Olympiad Level 2, 3

Tags: geometry , Circumcenter , Locus , circle

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

2003 Argentina National Olympiad, 4

Tags: geometry , trapezoid , right triangle , Circumcenter , area of a triangle

The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.

Kharkiv City MO Seniors - geometry, 2016.11.5

Tags: geometry , Circumcenter , circles , equal segments , Kharkiv

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

2015 IMO Shortlist, G2

Tags: geometry , IMO Shortlist , IMO , IMO 2015 , circumcircle , Circumcenter , Triangle

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

2011 Sharygin Geometry Olympiad, 7

Tags: geometry , cyclic quadrilateral , Circumcenter , concurrent , circles , altitudes , circumcircle

Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1,B_1, C_1$ are the bases of its altitudes. Points $A', B', C'$ lying on lines $OA_1, OB_1, OC_1$ respectively are such that quadrilaterals $AOBC', BOCA', COAB'$ are cyclic. Prove that the circumcircles of triangles $AA_1A', BB_1B', CC_1C'$ have a common point.

2015 Kyiv Math Festival, P4

Tags: Kyiv mathematical festival , geometry , perimeter , Circumcenter

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 $DEF.$

2012 Junior Balkan Team Selection Tests - Romania, 4

Tags: cyclic quadrilateral , geometry , Concyclic , collinear , Circumcenter , circumcircle

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$. a) Prove that the points $P, Q$, and $R$ are collinear. b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.

2018 Thailand TST, 4

Tags: geometry , rectangle , colinear , orthocenter , Circumcenter

Let $\vartriangle ABC$ be an acute triangle with altitudes $AA_1, BB_1, CC_1$ and orthocenter $H$. Let $K, L$ be the midpoints of $BC_1, CB_1$. Let $\ell_A$ be the external angle bisector of $\angle BAC$. Let $\ell_B, \ell_C$ be the lines through $B, C$ perpendicular to $\ell_A$. Let $\ell_H$ be the line through $H$ parallel to $\ell_A$. Prove that the centers of the circumcircles of $\vartriangle A_1B_1C_1, \vartriangle AKL$ and the rectangle formed by $\ell_A, \ell_B, \ell_C, \ell_H$ lie on the same line.

2022 Bulgarian Spring Math Competition, Problem 9.2

Tags: geometry , median , Circumcenter

Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.

2019 Oral Moscow Geometry Olympiad, 3

Tags: geometry , circumcircle , Concyclic , Circumcenter , 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.

2023 Brazil EGMO Team Selection Test, 1

Tags: geometry , circumcircle , orthocenter , Circumcenter , Nine Point Circle

Let $\Delta ABC$ be a triangle with orthocenter $H$ and $\Gamma$ be the circumcircle of $\Delta ABC$ with center $O$. Consider $N$ the center of the circle that passes through the feet of the heights of $\Delta ABC$ and $P$ the intersection of the line $AN$ with the circle $\Gamma$. Suppose that the line $AP$ is perpendicular to the line $OH$. Prove that $P$ belongs to the reflection of the line $OH$ by the line $BC$.

2018 Czech-Polish-Slovak Junior Match, 4

Tags: geometry , circumcircle , colinear , 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 $AB$.

2005 Sharygin Geometry Olympiad, 16

Tags: geometry , angles , orthocenter , incenter , Circumcenter , Centroid

We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle

2005 Sharygin Geometry Olympiad, 5

Tags: geometry , Locus , orthocenter , Circumcenter , Centroid , circumcircle

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

2017 Ukrainian Geometry Olympiad, 4

Tags: geometry , parallelogram , circumcircle , Circumcenter , right angle

Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.

2000 Estonia National Olympiad, 4

Tags: geometry , similar triangles , Circumcenter , similar

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

2023 Junior Balkan Mathematical Olympiad, 4

Tags: geometry , circumcircle , Circumcenter

Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic. [i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]

1989 All Soviet Union Mathematical Olympiad, 504

Tags: geometry , Circumcenter , equidistant , angle bisector

$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.

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 $|AP| = |CQ|$.