Olympiad problems

2004 Mediterranean Mathematics Olympiad, 2

In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that \[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]

2008 China Team Selection Test, 1

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

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

2004 IMO Shortlist, 8

Tags: polars , brocard , power of a point , projective geometry , geometry , circumcircle

Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$. [i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]

2011 Polish MO Finals, 2

Tags: circumcircle , 3d geometry , geometry , tetrahedron

In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.

2005 Greece National Olympiad, 4

Tags: circumcircle , geometry

Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.

2021 Middle European Mathematical Olympiad, 3

Tags: concyclic , ptolemy sinus lemma , computer problems , geometry , circumcircle

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.

JBMO Geometry Collection, 1999

Tags: geometry , perpendicular bisector , trigonometry , circumcircle

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

Croatia MO (HMO) - geometry, 2014.7

Tags: incenter , circumcircle , geometry , tangent

Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.

Kvant 2024, M2817

Tags: geometry , circumcircle

We are given fixed circles $\Omega$ and $\omega$ such that there exists a hexagon $ABCDEF$ inscribed in $\Omega$ and circumscribed around $\omega$. (Note that then, by virtue of Poncelet's theorem, there is an infinite family of such hexagons.) Prove that the value of $\dfrac{S_{ABCDEF}}{AD+BE+CF}$ it does not depend on the choice of the hexagon $ABCDEF$. [i]A. Zaslavsky and Tran Quang Hung[/i]

2014 Postal Coaching, 2

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

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Estonia Open Senior - geometry, 2011.1.3

Tags: geometry , circumcircle , geometric inequality , perpendicular

Consider an acute-angled triangle $ABC$ and its circumcircle. Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.

2005 India IMO Training Camp, 1

Tags: circumcircle , euler , incenter , geometry

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

2005 Slovenia Team Selection Test, 5

Tags: circumcircle , homothety , triangle , geometry

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

1997 Pre-Preparation Course Examination, 2

Tags: parallelogram , ratio , incenter , rectangle , trapezoid , circumcircle , geometry

Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that: [b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point. [b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point. [b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.

2012 Germany Team Selection Test, 2

Tags: circumcircle , geometry , 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]

1999 AMC 12/AHSME, 29

Tags: circumcircle , tetrahedron , 3d geometry , geometry , inradius , sphere , probability

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 0.1\qquad \textbf{(C)}\ 0.2\qquad \textbf{(D)}\ 0.3\qquad \textbf{(E)}\ 0.4$

2013 Turkey Team Selection Test, 3

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

Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]

Russian TST 2021, P3

Tags: geometry , circumcircle , incenter , tangent

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.

2009 Indonesia TST, 3

Tags: circumcircle , geometry

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

2022 Taiwan TST Round 3, 5

Tags: concyclic , geometry , circumcircle

Let $ABC$ be an acute triangle with circumcenter $O$ and circumcircle $\Omega$. Choose points $D, E$ from sides $AB, AC$, respectively, and let $\ell$ be the line passing through $A$ and perpendicular to $DE$. Let $\ell$ intersect the circumcircle of triangle $ADE$ and $\Omega$ again at points $P, Q$, respectively. Let $N$ be the intersection of $OQ$ and $BC$, $S$ be the intersection of $OP$ and $DE$, and $W$ be the orthocenter of triangle $SAO$. Prove that the points $S$, $N$, $O$, $W$ are concyclic. [i]Proposed by Li4 and me.[/i]

2007 Bosnia Herzegovina Team Selection Test, 5

Tags: circumcircle , geometry , angle bisector

Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN$. Let $T$ be the intersection point of angle bisector of $\angle ABC$ and line from $K$ perpendicular to $AN$. Determine value of $\angle MTB$.

1992 India Regional Mathematical Olympiad, 4

Tags: geometry , parallelogram , rectangle , circumcircle , power of a point , trigonometry , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

2002 Mexico National Olympiad, 2

Tags: parallelogram , circumcenter , circumcircle , geometry

$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$. The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$. Show that the circumcenter of $CEF$ lies on $K$.

2024 Sharygin Geometry Olympiad, 21

Tags: geometry , circumcircle

A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC, AC$ at points $A', B'$ respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. Prove that the lines $AA', BB'$ and $CC'$ concur.

2020 CHKMO, 3

Tags: circumcircle , incircle , geometry

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 $\angle PBI=90^{\circ}$.