Olympiad problems

2001 Korea - Final Round, 2

In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.

2009 Sharygin Geometry Olympiad, 12

Tags: reflection , law of sines , trig identities , geometric transformation , circumcircle , trigonometry , geometry

Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.

2017 Korea Junior Math Olympiad, 2

Tags: intersection , circumcircle , geometry

Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

2019 Irish Math Olympiad, 8

Tags: concurrency , geometry , circumcircle , parallelogram

Consider a point $G$ in the interior of a parallelogram $ABCD$. A circle $\Gamma$ through $A$ and $G$ intersects the sides $AB$ and $AD$ for the second time at the points $E$ and $F$ respectively. The line $FG$ extended intersects the side $BC$ at $H$ and the line $EG$ extended intersects the side $CD$ at $I$. The circumcircle of triangle $HGI$ intersects the circle $\Gamma$ for the second time at $M \ne G$. Prove that $M$ lies on the diagonal $AC$.

2019 Balkan MO Shortlist, G3

Tags: tangent , circumcircle , geometry

Let $ABC$ be a scalene and acute triangle with circumcenter $O$. Let $\omega$ be the circle with center $A$, tangent to $BC$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $FG \perp AO$, $\angle BFD = \angle DGC$ and the couples of points $(B,F)$ and $(C,G)$ are in different halfplanes with respect to the line $AD$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $ABC$.

2010 Switzerland - Final Round, 9

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

Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

2014 Taiwan TST Round 2, 1

Tags: reflection , circumcircle , incenter , geometry

Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic. [i]Proposed by Telv Cohl[/i]

2014 Iran Team Selection Test, 1

Tags: circumcircle , geometry

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.

2009 IMO Shortlist, 8

Tags: geometry , circumcircle , trigonometry , inradius , incenter

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2023 USA EGMO Team Selection Test, 3

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

2019 Sharygin Geometry Olympiad, 6

Tags: angle chasing , circumcircle , geometry , angle bisector , median

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.

2009 Sharygin Geometry Olympiad, 10

Tags: circumcircle , geometry , trapezoid , perpendicular bisector

Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$.

2010 CentroAmerican, 2

Tags: circumcircle , geometry , projective geometry

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 $BQ$.

2022 Macedonian Mathematical Olympiad, Problem 5

Tags: excenter , collinearity , circumcircle , geometry , incenter

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$. $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$. The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$. Prove that $A', D$ and $Z$ are collinear. [i]Proposed by Nikola Velov[/i]

2019 China National Olympiad, 3

Tags: geometry , circumcircle

Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.

2013 China Team Selection Test, 1

Tags: reflection , circumcircle , geometry , geometric transformation

The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.

2006 Romania Team Selection Test, 4

Tags: parallelogram , trigonometry , geometry , incenter , circumcircle

Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$. Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.

2013 Oral Moscow Geometry Olympiad, 4

Tags: concurrency , circumcircle , geometry , angle bisector , equal segments

Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.

2022 APMO, 2

Tags: circumcircle , geometry

Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.

2023 Philippine MO, 3

Tags: geometry , circumcircle

In $\triangle ABC$, $AB > AC$. Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$, and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$. Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$. The line through $B$ parallel to $QN$ meets $PN$ at $D$, and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$. Show that the lines $PM$, $BE$, and $AC$ are concurrent.

2021 Iran MO (3rd Round), 3

Tags: circumcircle , geometry

Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).

1996 China National Olympiad, 1

Tags: similar triangles , trigonometry , conic , circumcircle , geometry

Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.

2019 Bulgaria EGMO TST, 2

Tags: circumcircle , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ centered at $O$, whose diagonals intersect at $H$. Let $O_1$ and $O_2$ be the circumcenters of triangles $AHD$ and $BHC$. A line through $H$ intersects $\omega$ at $M_1$ and $M_2$ and intersects the circumcircles of triangles $O_1HO$ and $O_2HO$ at $N_1$ and $N_2$, respectively, so that $N_1$ and $N_2$ lie inside $\omega$. Prove that $M_1N_1 = M_2N_2$.

2018 Junior Balkan MO, 4

Tags: circumcircle , geometry

Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.

2018 Yasinsky Geometry Olympiad, 6

Tags: perpendicular , isosceles triangle , geometry , perpendicularity , circumcircle , incenter , isosceles

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)