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$.

2013 Romania Team Selection Test, 2

Tags: incenter , circumcircle , geometric transformation , homothety , power of a point , geometry

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

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]

2011 ELMO Shortlist, 6

Tags: algebra , polynomial , counting , derangement , induction , geometry , geometric transformation

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

2007 Hungary-Israel Binational, 2

Tags: ellipse , geometry , 3d geometry , sphere , geometric transformation , rotation , conic

Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.

2013 Korea - Final Round, 4

Tags: circumcircle , geometric transformation , homothety , geometry

For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.

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?

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 $\angle BAD+\angle ADC = 120$.

2013 China Team Selection Test, 1

Tags: modular arithmetic , geometric transformation , real analysis , number theory , geometry

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

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 .

2010 USA Team Selection Test, 4

Tags: geometry , incenter , geometric transformation , homothety

Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.

2012 EGMO, 7

Tags: geometry , circumcircle , geometric transformation

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.) [i]Luxembourg (Pierre Haas)[/i]

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.

1997 USAMO, 4

Tags: geometry , modular arithmetic , geometric transformation , homothety , combinatorial geometry

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

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.

1993 All-Russian Olympiad, 2

Tags: geometry , rectangle , geometric transformation , rotation , symmetry , combinatorics

Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?