Olympiad problems

2013 India Regional Mathematical Olympiad, 5

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

In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

2005 Harvard-MIT Mathematics Tournament, 10

Tags: circumcircle , geometric transformation , dilation , radical axis , power of a point , reflection , geometry

Let $AB$ be a diameter of a semicircle $\Gamma$. Two circles, $\omega_1$ and $\omega_2$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $AB$ at $P$ and $Q$, respectively, and to semicircular arc $AB$ at $C$ and $D$, respectively, with $AP<AQ$. Suppose $F$ lies on $\Gamma$ such that $ \angle FQB = \angle CQA $ and that $ \angle ABF = 80^\circ $. Find $ \angle PDQ $ in degrees.

2012 Sharygin Geometry Olympiad, 9

Tags: reflection , geometry , geometric transformation

In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.

2022 Turkey EGMO TST, 1

Tags: circumcircle , reflection , geometry , geometric transformation

Given an acute angle triangle $ABC$ with circumcircle $\Gamma$ and circumcenter $O$. A point $P$ is taken on the line $BC$ but not on $[BC]$. Let $K$ be the reflection of the second intersection of the line $AP$ and $\Gamma$ with respect to $OP$. If $M$ is the intersection of the lines $AK$ and $OP$, prove that $\angle OMB+\angle OMC=180^{\circ}$.

1979 USAMO, 2

Tags: 3d geometry , geometry , reflection , angle bisector , sphere , geometric transformation

Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$. [b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.

Cono Sur Shortlist - geometry, 2005.G3.4

Tags: perpendicular bisector , reflection , symmetry , geometry , geometric transformation , incenter

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

2014 Middle European Mathematical Olympiad, 6

Tags: geometric transformation , reflection , geometry , homothety , perpendicular bisector

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B, C, N,$ and $L$ are concyclic.

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 $PQ \perp CD$.

2014 Bundeswettbewerb Mathematik, 2

Tags: 3d geometry , algorithm , induction , geometry , reflection , prism , geometric transformation

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

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.

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 $P$.

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.

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 $ABC.$

2009 China Team Selection Test, 2

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

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

2015 Tournament of Towns, 2

Tags: reflection , circumcircle , geometric transformation , geometry

A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

2005 Polish MO Finals, 2

Tags: symmetry , reflection , trapezoid , angle bisector , geometric transformation , geometry , circumcircle

The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.

2022 Sharygin Geometry Olympiad, 8.5

Tags: reflection , geometric transformation , geometry

An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$ B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.

2008 Vietnam Team Selection Test, 1

Tags: ratio , perpendicular bisector , geometry , reflection , angle bisector , geometric transformation

On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$. $ 1.$ Prove that $ K$ always lie on a fixed line. $ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.

2013 USA TSTST, 3

Tags: reflection , slope , analytic geometry , floor function , graphing lines , geometric transformation

Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.

2008 China Western Mathematical Olympiad, 1

Tags: analytic geometry , reflection , geometric transformation , number theory , geometry

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

2008 China Team Selection Test, 1

Tags: circumcircle , geometry , reflection , geometric transformation

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$.

2008 National Olympiad First Round, 17

Tags: geometry , geometric transformation , pythagorean theorem , reflection

Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac 52 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2014 PUMaC Geometry B, 5

Tags: trig identities , geometry , geometric transformation , perpendicular bisector , reflection , trigonometry , cyclic quadrilateral

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

2008 ISI B.Stat Entrance Exam, 10

Tags: topology , reflection , geometry , homothety , function , geometric transformation , algebra

Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto. (i) Show that any two line segments in the plane are equivalent. (ii) Show that any two circles in the plane are equivalent.

2000 Bulgaria National Olympiad, 2

Tags: geometric transformation , reflection , geometry

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.