Olympiad problems

2019 Oral Moscow Geometry Olympiad, 4

The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

2011 Belarus Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

1977 IMO Longlists, 56

Tags: 3d geometry , tetrahedron , circumcircle , geometry

The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.

Geometry Mathley 2011-12, 2.2

Tags: geometry , circumcircle , circles , concurrency

Let $ABC$ be a scalene triangle. A circle $(O)$ passes through $B,C$, intersecting the line segments $BA,CA$ at $F,E$ respectively. The circumcircle of triangle $ABE$ meets the line $CF$ at two points $M,N$ such that $M$ is between $C$ and $F$. The circumcircle of triangle $ACF$ meets the line $BE$ at two points $P,Q$ such that $P$ is betweeen $B$ and $E$. The line through $N$ perpendicular to $AN$ meets $BE$ at $R$, the line through $Q$ perpendicular to $AQ$ meets $CF$ at $S$. Let $U$ be the intersection of $SP$ and $NR, V$ be the intersection of $RM$ and $QS$. Prove that three lines $NQ,UV$ and $RS$ are concurrent. Trần Quang Hùng

Kyiv City MO Juniors 2003+ geometry, 2011.9.41

Tags: tangent , geometry , circumcircle , equal segments

The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

2012 Grigore Moisil Intercounty, 4

Tags: complex number , circumcircle

[b]a)[/b] Let $ A $ denote the complex numbers of modulus $ 1/3, $ and $ B $ denote the complex numbers of modulus at least $ 1/2. $ Show that $ A+B=AB\neq\mathbb{C} . $ [b]b)[/b] Prove that there is no family $ Y $ of complex numbers that satisfies $ X+Y=XY\neq\mathbb{C} , $ where $ X $ denotes the complex numbers of modulus $ 1. $

1959 IMO, 5

Tags: geometry , circumcircle , perpendicular bisector , angle bisector

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$. a) Prove that $N$ and $N'$ coincide; b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$; c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

2000 Tournament Of Towns, 2

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

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

2008 China Team Selection Test, 1

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

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

2020 Thailand TSTST, 3

Tags: geometry , circumcircle , reflection , collinear

Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.

2003 Finnish National High School Mathematics Competition, 1

Tags: incenter , circumcircle , geometry

The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

2015 Sharygin Geometry Olympiad, 8

Tags: geometry , circumcircle

A perpendicular bisector of side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.

2006 Iran MO (3rd Round), 2

Tags: circumcircle , parallelogram , rectangle , geometric transformation , geometry

$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.

Cono Sur Shortlist - geometry, 2018.G2.5

Tags: geometry , incenter , circumcircle

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

2014 ELMO Shortlist, 11

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

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

2008 Harvard-MIT Mathematics Tournament, 32

Tags: trigonometry , circumcircle , symmetry , geometry

Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.

2005 Turkey Junior National Olympiad, 1

Tags: geometry , circumcircle , ratio

Let $ABC$ be an acute triangle. Let$H$ and $D$ be points on $[AC]$ and $[BC]$, respectively, such that $BH \perp AC$ and $HD \perp BC$. Let $O_1$ be the circumcenter of $\triangle ABH$, and $O_2$ be the circumcenter of $\triangle BHD$, and $O_3$ be the circumcenter of $\triangle HDC$. Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$.

2000 Federal Competition For Advanced Students, Part 2, 1

Tags: euler , circumcircle , angle bisector , geometry

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

1986 IMO Longlists, 31

Tags: circumcircle , ratio , geometry

Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , geometry , circumcircle

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2015 Oral Moscow Geometry Olympiad, 6

Tags: isosceles , altitude , circumcircle , circles , concurrency , geometry

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.

2009 Indonesia MO, 4

Tags: trigonometry , circumcircle , angle bisector , perpendicular bisector , geometry

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

2014 Vietnam National Olympiad, 4

Tags: circumcircle , parallelogram , geometric transformation , reflection , trapezoid , geometry

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$ a) Prove that $EF=\frac{BC}{2}.$ b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

2025 Olympic Revenge, 2

Tags: geometry , circumcircle

Let $ABC$ be a scalene triangle with $\Omega_A, \Omega_B,\Omega_C$ its excircles. $T_A$ is the intersection point of the external tangent (different of $AB$) of $\Omega_A,\Omega_B$ with the external tangent (different of $AC$) of $\Omega_A, \Omega_C$. Define $T_B, T_C$ in a similar way. If $I_A, I_B, I_C$ are the excenters of $ABC$, prove that the circumcircles of $AI_AT_A, BI_BT_B, CI_CT_C$ concur in exactly two points.

2018 Danube Mathematical Competition, 3

Tags: geometry , parallel , angle bisector , circumcircle , perpendicular bisector

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.