Olympiad problems

2011 Romanian Master of Mathematics, 3

Tags: geometry , geometric transformation , parabola , circumcircle , trigonometry , conic

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

2011 Turkey Team Selection Test, 1

Tags: circumcircle , inequalities , geometric transformation , geometry , homothety , ratio

Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that \[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]

2016 Flanders Math Olympiad, 1

Tags: circumcircle , isosceles , parallel , geometry

In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles. [img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]

2009 Ukraine Team Selection Test, 5

Tags: orthocenter , right triangle , circumcircle , circumcenter , geometry

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

2016 Sharygin Geometry Olympiad, P11

Tags: symmedian , construction , circumcircle , geometry

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

2021 Israel TST, 3

Tags: circumcircle , geometry , incenter

In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.

Indonesia Regional MO OSP SMA - geometry, 2017.3

Tags: symmetric , orthocenter , circumcircle , geometry

Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.

2004 District Olympiad, 2

Tags: circumcircle , analytic geometry , geometry

Find the possible coordinates of the vertices of a triangle of which we know that the coordinates of its orthocenter are $ (-3,10), $ those of its circumcenter is $ (-2,-3), $ and those of the midpoint of some side is $ (1,3). $

2006 Iran Team Selection Test, 5

Tags: circumcircle , parallelogram , angle bisector , geometry , trigonometry

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

2004 All-Russian Olympiad, 4

Tags: geometry , symmetry , circumcircle , homothety , analytic geometry , geometric transformation

Let $O$ be the circumcenter of an acute-angled triangle $ABC$, let $T$ be the circumcenter of the triangle $AOC$, and let $M$ be the midpoint of the segment $AC$. We take a point $D$ on the side $AB$ and a point $E$ on the side $BC$ that satisfy $\angle BDM = \angle BEM = \angle ABC$. Show that the straight lines $BT$ and $DE$ are perpendicular.

2012 India National Olympiad, 1

Tags: inequalities , topology , circumcircle , geometry , trigonometry , function

Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.

2009 CentroAmerican, 5

Tags: geometric transformation , circumcircle , homothety , projective geometry , geometry

Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.

2024 Saint Petersburg Mathematical Olympiad, 6

Tags: circumcircle , geometry

Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.

2006 Poland - Second Round, 2

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

Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.

1994 Brazil National Olympiad, 2

Tags: trig identities , circumcircle , combinatorial geometry , law of sines , geometry , trigonometry

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.

2002 India IMO Training Camp, 4

Tags: orthocenter , triangle , concurrency , circumcircle , geometry

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

2008 China Team Selection Test, 1

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

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

2018 India Regional Mathematical Olympiad, 1

Tags: angle bisector , geometry , circumcircle

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

2025 Benelux, 3

Tags: incenter , circumcircle , geometry

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

2016 China Team Selection Test, 1

Tags: circumcircle , reflection , geometric transformation , geometry

$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.

2020 Candian MO, 3#

Tags: really old , geometry , circumcircle , ratio , algebra

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.

2006 Irish Math Olympiad, 2

Tags: geometry , circumcircle

$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$. Moreover, neither $P$ nor $Q$ is a vertex of $ABC$. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$.

2013 Ukraine Team Selection Test, 8

Tags: triangle , reflection , trigonometry , geometry , circumcircle

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

2010 Contests, 1

Tags: rhombus , parallelogram , geometry , incenter , circumcircle , ratio

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

1967 IMO Longlists, 41

Tags: triangle , concurrency , reflection , circumcircle , geometry

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$