Olympiad problems

2006 QEDMO 3rd, 1

Tags: circumcircle , geometry

2015 Greece Team Selection Test, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

2007 Peru IMO TST, 2

Tags: circumcircle , trigonometry , ratio , rectangle , analytic geometry , geometric transformation , geometry

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

2011 IMO Shortlist, 1

Tags: geometry , circumcircle , trigonometry , geometric transformation

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

2014 Saudi Arabia IMO TST, 4

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , perpendicular bisector , angle bisector

Points $A_1,~ B_1,~ C_1$ lie on the sides $BC,~ AC$ and $AB$ of a triangle $ABC$, respectively, such that $AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1$. Let $I_A,~ I_B,~ I_C$ be the incenters of triangles $AB_1C_1,~ A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcenter of triangle $I_AI_BI_C$, is the incenter of triangle $ABC$.

2009 IMO, 2

Tags: geometry , circumcircle , reflection

Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$ [i]Proposed by Sergei Berlov, Russia [/i]

2010 IberoAmerican, 2

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

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

2012 Benelux, 3

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

In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.

1997 IMO Shortlist, 20

Tags: geometry , circumcircle , cyclic quadrilateral , triangle

A quick solution: Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.

2011 Turkey MO (2nd round), 2

Tags: circumcircle , trigonometry , geometric transformation , reflection , perpendicular bisector , cyclic quadrilateral , geometry

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

2021 Saudi Arabia Training Tests, 17

Tags: geometry , circumcircle , tangent circle

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Tangent lines to $(O)$ at $B,C$ meet at $T$. A line passes through $T$ cuts segments $AB$ at $D$ and cuts ray $CA$ at $E$. Take $M$ as midpoint of $DE$ and suppose that $MA$ cuts $(O)$ again at $K$. Prove that $(MKT)$ is tangent to $(O)$.

2011 Morocco National Olympiad, 3

Tags: geometric transformation , homothety , search , function , circumcircle , power of a point , geometry

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.

1996 Brazil National Olympiad, 4

Tags: geometry , circumcircle , vector , ratio , trigonometry , perpendicular bisector , geometric transformation

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

2014 India Regional Mathematical Olympiad, 1

Tags: circumcircle , incenter , perpendicular bisector , geometry

In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

1995 Italy TST, 4

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

In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.

2006 Baltic Way, 15

Tags: circumcircle , trapezoid , geometry

Let the medians of the triangle $ABC$ intersect at point $M$. A line $t$ through $M$ intersects the circumcircle of $ABC$ at $X$ and $Y$ so that $A$ and $C$ lie on the same side of $t$. Prove that $BX\cdot BY=AX\cdot AY+CX\cdot CY$.

2010 Turkey MO (2nd round), 2

Tags: circumcircle , ratio , geometry

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

Brazil L2 Finals (OBM) - geometry, 2013.3

Tags: midpoint , projection , geometry , circumcircle , perpendicular

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

2023 Yasinsky Geometry Olympiad, 6

Tags: geometry , circles , circumcircle

In the triangle $ABC$ with sides $AC = b$ and $AB = c$, the extension of the bisector of angle $A$ intersects it's circumcircle at point with $W$. Circle $\omega$ with center at $W$ and radius $WA$ intersects lines $AC$ and $AB$ at points $D$ and $F$, respectively. Calculate the lengths of segments $CD$ and $BF$. (Evgeny Svistunov) [img]https://cdn.artofproblemsolving.com/attachments/7/e/3b340afc4b94649992eb2dccda50ca8f3f7d1d.png[/img]

2023 JBMO Shortlist, G3

Tags: geometry , circumcircle , tangent circle

Let $A,B,C,D$ and $E$ be five points lying in this order on a circle, such that $AD=BC$. The lines $AD$ and $BC$ meet at a point $F$. The circumcircles of the triangles $CEF$ and $ABF$ meet again at the point $P$. Prove that the circumcircles of triangles $BDF$ and $BEP$ are tangent to each other.

2015 Irish Math Olympiad, 8

Tags: geometry , perpendicular , circumcircle

In triangle $\triangle ABC$, the angle $\angle BAC$ is less than $90^o$. The perpendiculars from $C$ on $AB$ and from $B$ on $AC$ intersect the circumcircle of $\triangle ABC$ again at $D$ and $E$ respectively. If $|DE| =|BC|$, find the measure of the angle $\angle BAC$.

2013 AIME Problems, 8

Tags: trapezoid , quadratic , trigonometry , circumcircle , algebra , geometry

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

2019 Belarusian National Olympiad, 11.4

Tags: geometry , circumcircle

The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$. Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$. [i](A. Voidelevich)[/i]

1986 IMO Longlists, 44

Tags: incenter , circumcircle , collinearity , geometry

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2020 CHKMO, 3

Tags: geometry , incircle , circumcircle

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.