Olympiad problems

2001 IMO Shortlist, 7

Let $O$ be an interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.

2006 Victor Vâlcovici, 2

Tags: circumcircle , analytic geometry , geometry

Consider a point $ B $ on a segment $ AC. $ Find the locus of the points $ M $ that have the property that the circumcircles of $ ABM $ and $ BCM $ have equal radii. [i]Nicolae Soare[/i]

2019 Dutch IMO TST, 3

Tags: parallelogram , geometry , circumcircle , concurrency

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

2008 Peru Iberoamerican Team Selection Test, P2

Tags: circumcircle , homothety , trigonometry , geometry , quadrilateral , inversion

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2009 Belarus Team Selection Test, 2

Tags: circumcircle , trapezoid , geometry

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

2000 Singapore Team Selection Test, 1

Tags: angle bisector , circumcircle , geometry

In a triangle $ABC$, $AB > AC$, the external bisector of angle $A$ meets the circumcircle of triangle $ABC$ at $E$, and $F$ is the foot of the perpendicular from $E$ onto $AB$. Prove that $2AF = AB - AC$

Geometry Mathley 2011-12, 15.3

Tags: perpendicular bisector , circumcircle , geometry , symmedian

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

2014 China Team Selection Test, 1

Tags: geometric transformation , homothety , cyclic quadrilateral , trigonometry , circumcircle , geometry

Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

Brazil L2 Finals (OBM) - geometry, 2020.5

Tags: geometry , circumcircle

Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.

2012 Iran Team Selection Test, 2

Tags: angle bisector , circumcircle , trigonometry , power of a point , geometry , incenter , config geo

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]

2011 China Girls Math Olympiad, 2

Tags: perpendicular bisector , rectangle , circumcircle , geometry , trapezoid , projective geometry , geometric transformation

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

2013 Irish Math Olympiad, 5

Tags: circles , circumcircle , geometry

$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.

2016 Costa Rica - Final Round, G1

Tags: circumcircle , geometry

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

2011 Romania Team Selection Test, 2

Tags: search , circumcircle , geometry , euler , incenter , homothety , geometric transformation

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

2023 Germany Team Selection Test, 1

Tags: perpendicular bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.

2002 Iran MO (2nd round), 4

Tags: circumcircle , geometry

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

2024 Kyiv City MO Round 2, Problem 3

Tags: geometry , circumcircle

Let $\omega$ denote the circumscribed circle of an acute-angled $\triangle ABC$ with $AB \neq BC$. Let $A'$ be the point symmetric to the point $A$ with respect to the line $BC$. The lines $AA'$ and $A'C$ intersect $\omega$ for the second time at points $D$ and $E$, respectively. Let the lines $AE$ and $BD$ intersect at point $P$. Prove that the line $A'P$ is tangent to the circumscribed circle of $\triangle A'BC$. [i]Proposed by Oleksii Masalitin[/i]

2013 ELMO Shortlist, 12

Tags: geometric transformation , circumcircle , ratio , reflection , geometry

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

2011 Puerto Rico Team Selection Test, 4

Tags: geometry , circumcircle , power of a point , radical axis

Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

1985 IMO, 1

Tags: circumcircle , trigonometry , cyclic quadrilateral , geometry

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

2021 Peru Cono Sur TST., P3

Tags: circumcircle , geometry

Let $ABC$ be a triangle and $D$ is a point in $BC$. The line $DA$ cuts the circumcircle of $ABC$ in the point $E$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $F=MN\cap AD$ and $G\neq F$ is the point of intersection of the circumcircles of $\triangle DNF$ and $\triangle ECF$. Prove that $B,F$ and $G$ are collinears.

2013 Junior Balkan MO, 2

Tags: analytic geometry , circumcircle , trigonometry , geometry , graphing lines , slope , euler

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

2002 China Western Mathematical Olympiad, 1

Tags: trapezoid , circumcircle , geometry

Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.

2008 All-Russian Olympiad, 3

Tags: analytic geometry , circumcircle , trapezoid , trigonometry , geometry , parallelogram , power of a point

A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.

2012 Iran MO (3rd Round), 1

Tags: circumcircle , trigonometry , parallelogram , ratio , geometry

Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. [i]Proposed by Mehdi E'tesami Fard[/i]