Olympiad problems

2016 Croatia Team Selection Test, Problem 3

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

2015 Belarus Team Selection Test, 2

Tags: equal segments , geometry , circumcenter , cyclic quadrilateral

Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$. N.Sedrakian

2021 Baltic Way, 14

Tags: geometry , circumcenter , harmonic quadrilateral , symmedian , circumcircle

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.

1999 Tournament Of Towns, 2

Tags: concyclic , cyclic , circumcenter , circumcircle , geometry

Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with center $O$. Let $F$ be the second intersection point of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the intersection point of the segments $AC$ and $BD$. (A Zaslavskiy)

2018 Czech-Polish-Slovak Junior Match, 4

Tags: collinear , circumcenter , circumcircle , geometry

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 $AB$.

2009 Postal Coaching, 3

Tags: concyclic , circumcenter , altitude , geometry , incenter

Let $ABC$ be a triangle with circumcentre $O$ and incentre $I$ such that $O$ is different from $I$. Let $AK, BL, CM$ be the altitudes of $ABC$, let $U, V , W$ be the mid-points of $AK, BL, CM$ respectively. Let $D, E, F$ be the points at which the in-circle of $ABC$ respectively touches the sides $BC, CA, AB$. Prove that the lines $UD, VE, WF$ and $OI$ are concurrent.

2021 Caucasus Mathematical Olympiad, 7

Tags: orthocenter , circumcenter , geometry

An acute triangle $ABC$ is given. Let $AD$ be its altitude, let $H$ and $O$ be its orthocenter and its circumcenter, respectively. Let $K$ be the point on the segment $AH$ with $AK=HD$; let $L$ be the point on the segment $CD$ with $CL=DB$. Prove that line $KL$ passes through $O$.

Ukraine Correspondence MO - geometry, 2018.9

Tags: geometry , circumcenter , orthocenter

Let $ABC$ be an acute-angled triangle in which $AB <AC$. On the side $BC$ mark a point $D$ such that $AD = AB$, and on the side $AB$ mark a point $E$ such that the segment $DE$ passes through the orthocenter of triangle $ABC$. Prove that the center of the circumcircle of triangle $ADE$ lies on the segment $AC$.

2001 IMO, 1

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 $\angle A+\angle COP < 90^{\circ}$.

2023 Junior Balkan Mathematical Olympiad, 4

Tags: circumcircle , circumcenter , geometry

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]

2004 Estonia Team Selection Test, 2

Tags: parallel , geometry , 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.

2013 BAMO, 3

Tags: congruent triangles , circumcenter , orthocenter , geometry

Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.

2004 Estonia Team Selection Test, 2

Tags: equal segments , circumcenter , parallel , geometry

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.

2004 Germany Team Selection Test, 2

Tags: geometry , circumcircle , circumcenter , triangle , excenter

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]

2018 Greece JBMO TST, 2

Tags: semicircle , circumcenter , circumcircle , geometry

Let $ABC$ be an acute triangle with $AB<AC<BC, c$ it's circumscribed circle and $D,E$ be the midpoints of $AB,AC$ respectively. With diameters the sides $AB,AC$, we draw semicircles, outer of the triangle, which are intersected by line $D$ at points $M$ and $N$ respectively. Lines $MB$ and $NC$ intersect the circumscribed circle at points $T,S$ respectively. Lines $MB$ and $NC$ intersect at point $H$. Prove that: a) point $H$ lies on the circumcircle of triangle $AMN$ b) lines $AH$ and $TS$ are perpedicular and their intersection, let it be $Z$, is the circimcenter of triangle $AMN$

2020 Ukrainian Geometry Olympiad - December, 5

Tags: equal segments , geometry , circumcenter

Let $O$ is the center of the circumcircle of the triangle $ABC$. We know that $AB =1$ and $AO = AC = 2$ . Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively such that $OD = OE$ and $BD =\sqrt2 EC$. Find $OD^2$.

2015 Sharygin Geometry Olympiad, P2

Tags: angle , circumcenter , orthocenter , incenter , geometry

Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.

2018 Yasinsky Geometry Olympiad, 6

Tags: geometry , midpoint , circumcenter , inradius , incenter

Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$. (Grigory Filippovsky)

Brazil L2 Finals (OBM) - geometry, 2008.5

Tags: angle , circumcenter , orthocenter , geometry

Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .

1940 Moscow Mathematical Olympiad, 068

Tags: triangle construction , geometric construction , circumcenter , geometry

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

2012 Estonia Team Selection Test, 4

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

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 $|AP| = |CQ|$.

2018 Hanoi Open Mathematics Competitions, 9

Tags: geometry , perpendicular , circumcenter , similar triangles

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

1997 Dutch Mathematical Olympiad, 5

Tags: circumcenter , symmetry , geometry

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$. a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

1990 IMO Longlists, 96

Tags: circumcircle , circumcenter , orthocenter , geometry

Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$

Kharkiv City MO Seniors - geometry, 2016.10.3

Tags: circumcircle , geometry , collinear , circumcenter

Let $AD$ be the bisector of an acute-angled triangle $ABC$. The circle circumscribed around the triangle $ABD$ intersects the straight line perpendicular to $AD$ that passes through point $B$, at point $E$. Point $O$ is the center of the circumscribed circle of triangle $ABC$. Prove that the points $A, O, E$ lie on the same line.