Olympiad problems

2022 Taiwan TST Round 2, 5

Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$. Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$. [i]Proposed by ltf0501.[/i]

2016 Saudi Arabia GMO TST, 1

Tags: circumcircle , tangent , geometry

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK$ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Prove the following assertions: a) If $A$ is the center of the circle $(IMN)$, then $BC$ is tangent to $(IMN)$. b) If $I$ is the midpoint of $BC$, then $BC$ is equal to $4$ times of the distance between the centers of two circles $(ABK)$ and $(ACH)$.

2019 Dutch BxMO TST, 2

Tags: angle bisector , circumcircle , geometry

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

2019 Greece Team Selection Test, 2

Tags: circumcircle , geometry

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

1992 IMO Longlists, 81

Tags: circumcenter , orthocenter , geometry , circumcircle

Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$

2023 India Regional Mathematical Olympiad, 6

Tags: circumcircle , geometry , cyclic quadrilateral

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.

2019 Middle European Mathematical Olympiad, 6

Tags: circumcircle , geometry

Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear. [i]Proposed by Dominik Burek, Poland[/i]

2018 lberoAmerican, 2

Tags: circumcircle , geometry

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.

2023 Turkey MO (2nd round), 2

Tags: concurrent lines , circumcircle , external tangent , tangent circle , tangent , geometry

Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.

2008 APMO, 1

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

Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

2011 All-Russian Olympiad Regional Round, 11.3

Tags: rectangle , geometry , circumcircle

Point $K$ lies on the circumcircle of a rectangle $ABCD$. Line $CK$ intersects line segment $AD$ at point $M$ so that $AM:MD=2$. $O$ is the center the rectangle. Prove that the centroid of triangle $OKD$ belongs to the circumcircle of triangle $COD$. (Author: V. Shmarov)

2021 Brazil Team Selection Test, 3

Tags: circumcircle , geometry

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

2014 China Second Round Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$.

1992 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , circumradius , circumcircle

Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

2021 Middle European Mathematical Olympiad, 5

Tags: circumcircle , menelaus theorem , geometry

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

2010 Contests, 4

Tags: circumcircle , geometry

The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.

2008 Iran MO (3rd Round), 4

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

Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.

2014 Argentina Cono Sur TST, 5

Tags: geometry , angle bisector , circumcircle

In an acute triangle $ABC$, let $D$ be a point in $BC$ such that $AD$ is the angle bisector of $\angle{BAC}$. Let $E \neq B$ be the point of intersection of the circumcircle of triangle $ABD$ with the line perpendicular to $AD$ drawn through $B$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $E$, $O$, and $A$ are collinear.

2013 India IMO Training Camp, 3

Tags: circumcircle , geometry , perpendicular bisector , angle bisector

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

2009 USAMO, 5

Tags: inversion , reflection , homothety , geometric transformation , geometry , circumcircle

Trapezoid $ ABCD$, with $ \overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$. Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively. Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively. Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.

2024 Junior Balkan MO, 2

Tags: collinear , circumcircle , geometry

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

2012 Federal Competition For Advanced Students, Part 1, 4

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

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

2009 Sharygin Geometry Olympiad, 3

Tags: incircle , geometry , concurrency , circumcircle

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur. (I.Bogdanov)

2024 Israel TST, P2

Tags: circumcircle , mixtilinear incircle , excircle , incenter , geometry

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

2009 Benelux, 4

Tags: geometry , trapezoid , circumcircle

Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, let $E$ be a point on line $BC$ outside segment $BC$, such that segment $AE$ intersects segment $CD$. Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$. 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 $EF$, and assume that $K$ is different from $I$ and $J$. Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$.