Olympiad problems

2020 Ecuador NMO (OMEC), 5

In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$. The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$. Prove that $AB+AC>3BC$.

2012 Sharygin Geometry Olympiad, 12

Tags: incenter , circumcircle , geometry

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. A line passing through $O$ and parallel to $BC$ meets $AB$ and $AC$ in points $P$ and $Q$ respectively. The sum of distances from $O$ to $AB$ and $AC$ is equal to $OA$. Prove that $PB + QC = PQ$.

2005 National Olympiad First Round, 21

Tags: circumcircle , geometry

What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and midpoint of its side $[BC]$? $ \textbf{(A)}\ \dfrac {\sqrt 3}4 \qquad\textbf{(B)}\ \dfrac {\sqrt 5}4 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \dfrac {\sqrt {10}}4 $

2009 Germany Team Selection Test, 2

Tags: geometry , quadrilateral , inversion , trigonometry , homothety , circumcircle

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2009 China Western Mathematical Olympiad, 2

Tags: circumcircle , geometry

Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

2019 Philippine TST, 2

Tags: circumcircle , harmonic bundles , geometry

In a triangle $ABC$ with circumcircle $\Gamma$, $M$ is the midpoint of $BC$ and point $D$ lies on segment $MC$. Point $G$ lies on ray $\overrightarrow{BC}$ past $C$ such that $\frac{BC}{DC} = \frac{BG}{GC}$, and let $N$ be the midpoint of $DG$. The points $P$, $S$, and $T$ are defined as follows: [list = i] [*] Line $CA$ meets the circumcircle $\Gamma_1$ of triangle $AGD$ again at point $P$. [*] Line $PM$ meets $\Gamma_1$ again at $S$. [*] Line $PN$ meets the line through $A$ that is parallel to $BC$ at $Q$. Line $CQ$ meets $\Gamma$ again at $T$. [/list] Prove that the points $P$, $S$, $T$, and $C$ are concyclic.

2019 Ramnicean Hope, 2

Tags: geometry , circumcircle , vector geometry , median

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

2012 Germany Team Selection Test, 2

Tags: trigonometry , circumcircle , geometry , 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]

2012 Brazil Team Selection Test, 4

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

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2013 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2012 Romanian Master of Mathematics, 2

Tags: trigonometry , circumcircle , complex number , geometry

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

2015 Korea Junior Math Olympiad, 5

Tags: geometry , circumcircle , incenter

Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$. Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$. Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$. Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$. Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.

2023 Bulgaria National Olympiad, 2

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and $A_{1}, B_{1}, C_{1}$ be the touchpoints of the excircles with the segments $BC, CA, AB$ respectively. Let $O_{A}, O_{B}, O_{C}$ be the circumcenters of $\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}$ respectively. Prove that the lines through $O_{A}, O_{B}, O_{C}$ respectively parallel to the internal angle bisectors of $\angle A,\angle B, \angle C$ are concurrent.

2002 Iran MO (3rd Round), 5

Tags: radical axis , power of a point , circumcircle , 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 $MCS$.

2013 ELMO Shortlist, 2

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

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

1995 Iran MO (2nd round), 2

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

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 Iran MO (3rd round), 1

Tags: circumcircle , geometry

Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.

2011 Oral Moscow Geometry Olympiad, 6

Tags: altitude , geometry , circumcircle , concurrency

Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.

1992 IMO Longlists, 13

Tags: geometry , rhombus , convex quadrilateral , circumcircle

Let $ABCD$ be a convex quadrilateral such that $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$

2014 Online Math Open Problems, 29

Tags: 3d geometry , tetrahedron , geometry , calculus , circumcircle , integration

Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$. [i]Proposed by Sammy Luo and Evan Chen[/i]

2012 Polish MO Finals, 5

Tags: geometry , triangle , circumcircle , tangent

Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.

2019 Saudi Arabia Pre-TST + Training Tests, 3.1

Tags: circumcircle , perpendicular , geometry

In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.

OMMC POTM, 2022 12

Tags: circumcircle , geometry

Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$. [i]Proposed by Taco12[/i]

1969 IMO Shortlist, 44

Tags: radius , quadratic , geometry , circumcircle

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.

2002 China Team Selection Test, 2

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

Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$