As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$. [asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O); MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.

In the convex pentagon $ ABCDE$ following equalities holds: $ \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE$ and $ \angle ADB\equal{}\angle AEC$. The point $S$ is the intersection of the segments $BD$ and $CE$. Prove that lines $AS$ and $CD$ are perpendicular.

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$.

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$. [b](a)[/b] Prove that any two of the following statements imply the third. [list] [b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$. [b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$. [b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list] [b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that \[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC. \] Prove that \[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1. \]

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.

In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.

Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles? $\textbf{(A) }\dfrac{\sqrt3}4\qquad \textbf{(B) }\dfrac{\sqrt3}3\qquad \textbf{(C) }\dfrac23\qquad \textbf{(D) }\dfrac{\sqrt2}2\qquad \textbf{(E) }\dfrac{\sqrt3}2$

The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5, 1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?