Olympiad problems

2013 Online Math Open Problems, 30

Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$. [i]Ray Li[/i]

2003 Baltic Way, 14

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

Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

2000 Brazil National Olympiad, 1

Tags: reflection , geometry , geometric transformation

A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.

2013 Romanian Master of Mathematics, 6

Tags: geometry , reflection , geometric transformation , combinatorics

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

2018 Junior Balkan Team Selection Tests - Romania, 4

Tags: circumcircle , reflection , geometry

Let $ABC$ be a triangle, and let $E$ and $F$ be two arbitrary points on the sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. Let $D$ be the reflection of point $M$ across the line $EF$ and let $O$ be the circumcenter of triangle $ABC$. Prove that $D$ is on $BC$ if and only if $O$ belongs to the circumcircle of triangle $AEF$.

2021 Balkan MO Shortlist, G5

Tags: parallel , reflection , shortlist , geometry

Let $ABC$ be an acute triangle with $AC > AB$ and circumcircle $\Gamma$. The tangent from $A$ to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$. Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such that $MP$ is parallel to $AB$. Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$. [i]Proposed by Sam Bealing, United Kingdom[/i]

2005 Cono Sur Olympiad, 1

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

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 $r$.

2015 Mexico National Olympiad, 5

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

Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.

2002 France Team Selection Test, 1

Tags: reflection , geometry , circumcircle , geometric transformation

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

2013 Princeton University Math Competition, 5

Tags: reflection , graph theory , geometric transformation , geometry , rotation

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

1967 IMO Shortlist, 6

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

2011 Uzbekistan National Olympiad, 4

Tags: combinatorics , polynomial , reflection , geometry , algebra , geometric transformation

$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be $H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$ where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.

2022 Taiwan TST Round 3, G

Tags: geometry , geometric transformation , reflection , circumcircle

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. Let $M$ be the midpoint of side $BC$. Point $D$ is chosen from the minor arc $BC$ on $\Gamma$ such that $\angle BAD = \angle MAC$. Let $E$ be a point on $\Gamma$ such that $DE$ is perpendicular to $AM$, and $F$ be a point on line $BC$ such that $DF$ is perpendicular to $BC$. Lines $HF$ and $AM$ intersect at point $N$, and point $R$ is the reflection point of $H$ with respect to $N$. Prove that $\angle AER + \angle DFR = 180^\circ$. [i]Proposed by Li4.[/i]

2002 Dutch Mathematical Olympiad, 1

Tags: reflection , square , geometry

The sides of a $10$ by $10$ square $ABCD$ are reflective on the inside. A beam of light enters the square via the vertex $A$ and heads to the point $P$ on $CD$ with $CP = 3$ and $PD = 7$. In $P$ it naturally reflects on the $CD$ side. The light beam can only leave the square via one of the angular points $A, B, C$ or $D$. What is the distance that the light beam travels within the square before it leaves the square again? By which vertex does that happen?

2013 USAMO, 6

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

Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

2014 Tajikistan Team Selection Test, 2

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

Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

2008 Romania National Olympiad, 1

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

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

2019 Vietnam TST, P3

Tags: geometry , reflection , geometric transformation

Given an acute scalene triangle $ABC$ inscribed in circle $(O)$. Let $H$ be its orthocenter and $M$ be the midpoint of $BC$. Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$. Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$. a) Show that $AM$ bisects $D'X$. b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$. Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$. Let $G$ be the projection of the midpoint of $AS$ to the line $AO$. Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$.

2008 Ukraine Team Selection Test, 1

Tags: reflection , mixtilinear incircle , cyclic quadrilateral , geometry , circumcircle

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

2007 Ukraine Team Selection Test, 5

Tags: geometric transformation , euler , symmetry , geometry , parallelogram , circumcircle , reflection

$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

1994 Chile National Olympiad, 7

Tags: reflection , rectangle , geometry

Let $ABCD$ be a rectangle of length $m$ and width $n$, with $m, n$ positive integers. Consider a ray of light that starts from $A$, reflects with an angle of $45^o$ on an opposite side and continues reflecting away at the same angle. $\bullet$ For any pair $(m,n)$, show that the ray meets a vertex at some point. $\bullet$ Suppose $m$ and $n$ are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.

2020 Latvia Baltic Way TST, 10

Tags: reflection , geometric transformation , geometry

Given $\triangle ABC$ and it's orthocenter $H$. Point $P$ is arbitrary chosen on the side $ BC$. Let $Q$ and $R$ be reflections of point $P$ over sides $AB, AC$. It is given that points $Q,H,R$ are collinear. Prove that $\triangle ABC$ is right angled.

1965 AMC 12/AHSME, 4

Tags: reflection , geometric transformation , geometry

Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2007 Iran MO (3rd Round), 1

Tags: reflection , geometric transformation , parallelogram , geometry

Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.

2007 Iran Team Selection Test, 1

Tags: reflection , geometry , geometric transformation , analytic geometry

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]