Olympiad problems

2011 Singapore MO Open, 1

In the acute-angled non-isosceles triangle $ABC$, $O$ is its circumcenter, $H$ is its orthocenter and $AB>AC$. Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at the point $P$. Suppose $BD=DP$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. Prove that $\angle ODQ=90^{\circ}$.

2008 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.

2022 Azerbaijan JBMO TST, G3

Tags: geometry , geometric transformation , reflection

In acute, scalene Triangle $ABC$, $H$ is orthocenter,$ BD$ and $CE$ are heights. $X,Y$ are reflection of $A$ from $D$,$E$ respectively such that the points$ X,Y$ are on segments $DC$ and $EB$. The intersection of circles $ HXY$ and $ADE$ is $F.$ ( $F \neq H$). Prove that$ AF$ intersects middle point of $BC$. ( $M$ in the diagram is Midpoint of $BC$)

2002 India National Olympiad, 1

Tags: geometric transformation , reflection , geometry

For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements. ($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$. ($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$. ($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$. $ (a)$ Show that if all six of these statements are true then the hexagon is cyclic. $ (b)$ Prove that, in fact, five of the six statements suffice.

2008 India Regional Mathematical Olympiad, 1

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

Let $ ABC$ be an acute angled triangle; let $ D,F$ be the midpoints of $ BC,AB$ respectively. Let the perpendicular from $ F$ to $ AC$ and the perpendicular from $ B$ ti $ BC$ meet in $ N$: Prove that $ ND$ is the circumradius of $ ABC$. [15 points out of 100 for the 6 problems]

1996 USAMO, 5

Tags: isosceles triangle , trigonometry , law of cosines , law of sines , reflection , geometry

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

2012 India IMO Training Camp, 1

Tags: geometry , incenter , symmetry , reflection

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

2009 Princeton University Math Competition, 8

Tags: rotation , geometry , geometric transformation , reflection

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?

2010 Germany Team Selection Test, 2

Tags: incenter , reflection , inequalities , geometry

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

2007 Balkan MO Shortlist, G2

Tags: trigonometry , geometry , geometric transformation , reflection , trapezoid , circumcircle , trig identities

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

2008 Iran MO (3rd Round), 2

Tags: euler , geometric transformation , reflection , geometry

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

2009 Sharygin Geometry Olympiad, 5

Tags: geometric transformation , reflection , circumcircle , geometry

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

1997 Pre-Preparation Course Examination, 2

Tags: geometric transformation , reflection , geometry

Two circles $O, O'$ meet each other at points $A, B$. A line from $A$ intersects the circle $O$ at $C$ and the circle $O'$ at $D$ ($A$ is between $C$ and $D$). Let $M,N$ be the midpoints of the arcs $BC, BD$, respectively (not containing $A$), and let $K$ be the midpoint of the segment $CD$. Show that $\angle KMN = 90^\circ$.

2002 China Team Selection Test, 2

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

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}$

KoMaL A Problems 2017/2018, A. 705

Tags: geometry , geometric transformation , reflection

Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle. [i]Michael Ren[/i]

Indonesia MO Shortlist - geometry, g1

Tags: geometry , geometric transformation , reflection

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

2011 Belarus Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

2017 CentroAmerican, 1

Tags: geometry , geometric transformation , reflection , angle bisector

$ABC$ is a right-angled triangle, with $\angle ABC = 90^{\circ}$. $B'$ is the reflection of $B$ over $AC$. $M$ is the midpoint of $AC$. We choose $D$ on $\overrightarrow{BM}$, such that $BD = AC$. Prove that $B'C$ is the angle bisector of $\angle MB'D$. NOTE: An important condition not mentioned in the original problem is $AB<BC$. Otherwise, $\angle MB'D$ is not defined or $B'C$ is the external bisector.

2021 Iranian Geometry Olympiad, 4

Tags: geometry , reflection , trapezoid , concentric circles

In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric. [i]Proposed by Iman Maghsoudi - Iran[/i]

2008 China Team Selection Test, 1

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

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.

2020 Thailand TSTST, 3

Tags: geometry , circumcircle , reflection , collinear

Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.

2014 ELMO Shortlist, 11

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

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

2014 Vietnam National Olympiad, 4

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

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$ a) Prove that $EF=\frac{BC}{2}.$ b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

2001 USAMO, 6

Tags: reflection , geometry

Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

2005 Junior Balkan Team Selection Tests - Romania, 11

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

Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.