Olympiad problems

2010 CentroAmerican, 2

Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.

2002 Iran MO (3rd Round), 5

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

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.

2007 Polish MO Finals, 5

Tags: 3d geometry , tetrahedron , sphere , parallelogram , circumcircle , geometry

5. In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.

1998 USAMO, 2

Tags: geometry , circumcircle , perpendicular bisector , power of a point

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

2009 Hong Kong TST, 4

Tags: geometry , geometric transformation , homothety , ratio , circumcircle , trapezoid , projective geometry

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

2012 Kazakhstan National Olympiad, 2

Tags: circumcircle , geometry

Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that: $a)$ Points $M,N,A$ and $B$ lie on a circle $b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

2016 Saudi Arabia BMO TST, 2

Tags: geometry , excenter , circumcircle , concurrency

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

2022 Taiwan TST Round 2, 3

Tags: geometry , circumcircle , excircle , ddit

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

2019 Moroccan TST, 6

Tags: geometry , circumcircle

Let $ABC$ be a triangle. The tangent in $A$ of the circumcircle of $ABC$ cuts the line $(BC)$ in $X$. Let $A'$ be the symetric of $A$ by $X$ and $C'$ the symetric of $C$ by the line $(AX)$ Prove that the points $A, C', A'$ and $B$ are concyclic.

2013 ELMO Shortlist, 11

Tags: circumcircle , geometry

Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$. [i]Proposed by David Stoner[/i]

2020 Korea - Final Round, P5

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.

2004 Postal Coaching, 1

Tags: circumcircle , cyclic quadrilateral , geometry

Let $ABC$ and $DEF$ be two triangles such that $A+ D = 120^{\circ}$ and $B+E = 120^{\circ}$. Suppose they have the same circumradius. Prove that they have the same 'Fermat length'.

2008 India National Olympiad, 1

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

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

2000 Junior Balkan MO, 3

Tags: symmetry , circumcircle , angle bisector , projective geometry , cyclic quadrilateral , geometry

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

Kvant 2024, M2793

Tags: geometry , circumcircle , parallelogram , tangent circle

In acute triangle $ABC$ ($AB<AC$) point $O$ is center of its circumcircle $\Omega$. Let the tangent to $\Omega$ drawn at point $A$ intersect the line $BC$ at point $D$. Let the line $DO$ intersects the segments $AB$ and $AC$ at points $E$ and $F$, respectively. Point $G$ is constructed such that $AEGF$ is a parallelogram. Let $K$ and $H$ be points of intersection of segment $BC$ with segments $EG$ and $FG$, respectively. Prove that the circle $(GKH)$ touches the circle $\Omega$. [i] Proposed by Dong Luu [/i]

2000 Taiwan National Olympiad, 2

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

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

1997 IMO Shortlist, 18

Tags: geometry , circumcircle , trigonometry , triangle , easy geometry

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

2007 Federal Competition For Advanced Students, Part 2, 3

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

The triangle $ ABC$ with the circumcircle $ k(U,r)$ is given. On the extension of the radii $ UA$ a point $ P$ is chosen. The reflection of the line $ PB$ on the line $ BA$ is called $ g$. Likewise the reflection of the line $ PC$ on the line $ CA$ is called $ h$. The intersection of $ g$ and $ h$ is called $ Q$. Find the geometric location of all possible intersections $ Q$, while $ P$ passes through the extension of the radii $ UA$.

2019 IberoAmerican, 3

Tags: geometry , circumcircle

Let $\Gamma$ be the circumcircle of triangle $ABC$. The line parallel to $AC$ passing through $B$ meets $\Gamma$ at $D$ ($D\neq B$), and the line parallel to $AB$ passing through $C$ intersects $\Gamma$ to $E$ ($E\neq C$). Lines $AB$ and $CD$ meet at $P$, and lines $AC$ and $BE$ meet at $Q$. Let $M$ be the midpoint of $DE$. Line $AM$ meets $\Gamma$ at $Y$ ($Y\neq A$) and line $PQ$ at $J$. Line $PQ$ intersects the circumcircle of triangle $BCJ$ at $Z$ ($Z\neq J$). If lines $BQ$ and $CP$ meet each other at $X$, show that $X$ lies on the line $YZ$.

2006 MOP Homework, 1

Tags: circumcircle , geometry

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

2010 Tuymaada Olympiad, 2

Tags: inequalities , circumcircle , geometry

In acute triangle $ABC$, let $H$ denote its orthocenter and let $D$ be a point on side $BC$. Let $P$ be the point so that $ADPH$ is a parallelogram. Prove that $\angle DCP<\angle BHP$.

1995 APMO, 4

Tags: rectangle , circumcircle , parallelogram , cyclic quadrilateral , geometry

Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.

2009 Belarus Team Selection Test, 2

Tags: geometry , circumcircle , reflection

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

JBMO Geometry Collection, 2000

Tags: geometry , symmetry , circumcircle , angle bisector , projective geometry , cyclic quadrilateral

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

2001 Brazil Team Selection Test, Problem 4

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP : PQ : QC = AC : CB : BA$. Prove that the points $A$, $P$, $Q$ and $O$ lie on one circle. [i]Alternative formulation.[/i] Let $O$ be the center of the circumcircle of a triangle $ABC$. If $P$ and $Q$ are points on the sides $AB$ and $AC$, respectively, satisfying $\frac{BP}{PQ}=\frac{CA}{BC}$ and $\frac{CQ}{PQ}=\frac{AB}{BC}$, then show that the points $A$, $P$, $Q$ and $O$ lie on one circle.