Olympiad problems

2014 Taiwan TST Round 1, 3

Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.

2009 Sharygin Geometry Olympiad, 12

Tags: geometry , geometric transformation , reflection , circumcircle , trigonometry , trig identities , law of sines

Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.

2009 Iran MO (3rd Round), 1

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

1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.

2025 India STEMS Category A, 3

Tags: geometry , a-hm point , reflection

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

2010 Contests, 3

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

2014 China Team Selection Test, 4

Tags: geometry , circumcircle , geometric transformation , reflection , trigonometry , trig identities , law of sines

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

2008 ISI B.Stat Entrance Exam, 10

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

1995 Poland - First Round, 8

Tags: geometric transformation , reflection , geometry

The ray of light starts from the center of a square and reflects from its sides with the principle that the angle of reflection is equal to the angle of incidence. After some time the ray returns to the center of the square. The ray never reached the vertex and has never returned to the center of the square before. Prove that the ray reflected from the sides of the square an odd number of times.

2017 Baltic Way, 11

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

Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear. [i]Proposed by Mads Christensen, Denmark[/i]

2013 Online Math Open Problems, 26

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

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

2007 Kyiv Mathematical Festival, 4

Tags: geometric transformation , reflection , trapezoid , geometry

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

2011 Northern Summer Camp Of Mathematics, 3

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

Given an acute triangle $ABC$ such that $\angle C< \angle B< \angle A$. Let $I$ be the incenter of $ABC$. Let $M$ be the midpoint of the smaller arc $BC$, $N$ be the midpoint of the segment $BC$ and let $E$ be a point such that $NE=NI$. The line $ME$ intersects circumcircle of $ABC$ at $Q$ (different from $A, B$, and $C$). Prove that [b](i)[/b] The point $Q$ is on the smaller arc $AC$ of circumcircle of $ABC$. [b](ii)[/b] $BQ=AQ+CQ$

2008 Tuymaada Olympiad, 3

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

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

1967 IMO Longlists, 41

Tags: circumcircle , reflection , triangle , concurrency , 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 $ABC.$

2010 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , reflection , geometry

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

Cono Sur Shortlist - geometry, 2005.G3.4

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

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 $r$.

2008 China Western Mathematical Olympiad, 1

Tags: geometric transformation , reflection , analytic geometry , 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.

2011 Sharygin Geometry Olympiad, 8

Tags: geometry , incircle , concurrency , reflection

The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.

2019 Latvia Baltic Way TST, 12

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

Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

2022 China Second Round A2, 2

Tags: geometry , geometric transformation , reflection

$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.

2019 Philippine TST, 4

Tags: geometry , reflection , concyclic , parallelogram

Let $P$ be a point in parallelogram $ABCD$ such that $$PA \cdot PC + PB \cdot PD = AB \cdot BC.$$ Prove that the reflections of $P$ over lines $AB$, $BC$, $CD$, and $DA$ are concyclic.

2009 Tournament Of Towns, 4

Tags: geometric transformation , reflection , trigonometry , geometry

A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$ [i](4 points)[/i]

2012 NIMO Problems, 7

Tags: geometry , rectangle , geometric transformation , reflection

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$. [i]Proposed by Aaron Lin[/i]

2010 Iran MO (3rd Round), 6

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

In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points) the exam time was 4 hours and 30 minutes.

2014 Taiwan TST Round 2, 1

Tags: incenter , circumcircle , reflection , geometry

Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic. [i]Proposed by Telv Cohl[/i]