Olympiad problems

2020 Junior Balkan Team Selection Tests - Moldova, 11

Tags: circumcircle , geometry

Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle AFB$ is isosceles.

2013 National Olympiad First Round, 13

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

Let $D$ and $E$ be points on side $[BC]$ of a triangle $ABC$ with circumcenter $O$ such that $D$ is between $B$ and $E$, $|AD|=|DB|=6$, and $|AE|=|EC|=8$. If $I$ is the incenter of triangle $ADE$ and $|AI|=5$, then what is $|IO|$? $ \textbf{(A)}\ \dfrac {29}{5} \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ \dfrac {23}{5} \qquad\textbf{(D)}\ \dfrac {21}{5} \qquad\textbf{(E)}\ \text{None of above} $

2010 Indonesia TST, 4

Tags: trigonometry , geometry , geometric transformation , reflection , circumcircle , angle bisector , similar triangles

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

2013 National Olympiad First Round, 21

Tags: geometry , circumcircle , incenter , cyclic quadrilateral , angle bisector

Let $D$ and $E$ be points on side $[AB]$ of a right triangle with $m(\widehat{C})=90^\circ$ such that $|AD|=|AC|$ and $|BE|=|BC|$. Let $F$ be the second intersection point of the circumcircles of triangles $AEC$ and $BDC$. If $|CF|=2$, what is $|ED|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ 1+\sqrt 2 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 2\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

2002 Greece National Olympiad, 3

Tags: circumcircle , trigonometry , inequalities , function , geometry

In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$

2007 All-Russian Olympiad, 6

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

Let $ABC$ be an acute triangle. The points $M$ and $N$ are midpoints of $AB$ and $BC$ respectively, and $BH$ is an altitude of $ABC$. The circumcircles of $AHN$ and $CHM$ meet in $P$ where $P\ne H$. Prove that $PH$ passes through the midpoint of $MN$. [i]V. Filimonov[/i]

2007 All-Russian Olympiad, 4

Tags: circumcircle , asymptote , cyclic quadrilateral , angle bisector , geometry

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

2006 Poland - Second Round, 2

Tags: geometry , circumcircle , homothety , triangle -incenter

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

2017 Peru IMO TST, 9

Tags: geometry , circumcircle , cyclic quadrilateral , equal angles , tangent circle

Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the intersection point of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and $\omega$ are tangent to each other.

2006 Balkan MO, 2

Tags: geometry , circumcircle , projective geometry , algebra , polynomial , power of a point , radical axis

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point. [i]Greece[/i]

1997 Turkey MO (2nd round), 2

Tags: circumcircle , geometric transformation , geometry

In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$

2016 Sharygin Geometry Olympiad, P19

Tags: geometry , circumcircle , regular polygon , hexagon

Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.

2017 Romanian Master of Mathematics Shortlist, G1

Tags: geometry , trapezoid , circumcircle , moving points , conic , projective geometry

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

2012 Serbia Team Selection Test, 3

Tags: circumcircle , radical axis , geometry

Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$. a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic. b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

2010 Korea National Olympiad, 3

Tags: geometry , incenter , inradius , trigonometry , circumcircle , angle bisector , power of a point

Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.

2012 Iran Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , reflection , incenter , power of a point , radical axis

Let $O$ be the circumcenter of the acute triangle $ABC$. Suppose points $A',B'$ and $C'$ are on sides $BC,CA$ and $AB$ such that circumcircles of triangles $AB'C',BC'A'$ and $CA'B'$ pass through $O$. Let $\ell_a$ be the radical axis of the circle with center $B'$ and radius $B'C$ and circle with center $C'$ and radius $C'B$. Define $\ell_b$ and $\ell_c$ similarly. Prove that lines $\ell_a,\ell_b$ and $\ell_c$ form a triangle such that it's orthocenter coincides with orthocenter of triangle $ABC$. [i]Proposed by Mehdi E'tesami Fard[/i]

1995 Iran MO (2nd round), 2

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

Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$

2017 Peru IMO TST, 3

Tags: geometry , circumcircle , incircle , midpoint , tangent

The inscribed circle of the triangle $ABC$ is tangent to the sides $BC, AC$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ be the midpoint of $EF$. The circle circumscribed around the triangle $DMF$ intersects line $AB$ at $L$, the circle circumscribed around the triangle $DME$ intersects the line $AC$ at $K$. Prove that the circle circumscribed around the triangle $AKL$ is tangent to the line $BC$.

2011 Korea Junior Math Olympiad, 5

Tags: ratio , geometry , circumcircle , midpoint

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.

2013 China Team Selection Test, 2

Tags: circumcircle , geometric transformation , homothety , trigonometry , euler , geometry

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

2008 All-Russian Olympiad, 3

Tags: geometry , parallelogram , analytic geometry , trigonometry , trapezoid , power of a point , circumcircle

A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.

2010 Iran MO (3rd Round), 4

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

in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)

2018 China Northern MO, 6

Tags: geometry , circumcircle

Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$

2022 Taiwan TST Round 2, 5

Tags: geometry , circumcircle , miquel point , complete quadrilateral , tangent

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]

1990 Romania Team Selection Test, 2

Tags: function , geometry , circumcircle , inequalities

Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]