Reflect each of the shapes $A,B$ over some lines $l_A,l_B$ respectively and rotate the shape $C$ such that a $4 \times 4$ square is obtained. Identify the lines $l_A,l_B$ and the center of the rotation, and also draw the transformed versions of $A,B$ and $C$ under these operations. Proposed by Mahdi Etesamifard - Iran

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? Proposed by Mahdi Etesamifard - Iran

Inside a convex quadrilateral $ABCD$ with $BC>AD$, a point $T$ is chosen. $S$ lies on the segment $AT$ such that $DT = BC, \angle TSD = 90^\circ$. Prove that if $\angle DTA + \angle TAB + \angle ABC = 180^\circ$, then $AB + ST \geqslant CD + AS$. Proposed by Alexander Tereshin - Russia

An inscribed $n$-gon ($n > 3$), is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles obtained? (An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle) Proposed by Boris Frenkin - Russia

Points $Y,Z$ lie on the smaller arc $BC$ of the circumcircle of an acute triangle $\bigtriangleup ABC$ ($Y$ lies on the smaller arc $BZ$). Let $X$ be a point such that the triangles $\bigtriangleup ABC,\bigtriangleup XYZ$ are similar (in this exact order) with $A,X$ lying on the same side of $YZ$. Lines $XY,XZ$ intersect sides $AB,AC$ at points $E,F$ respectively. Let $K$ be the intersection of lines $BY,CZ$. Prove that one of the intersections of the circumcircles of triangles $\bigtriangleup AEF,\bigtriangleup KBC$ lie on the line $KX$. Proposed by Amirparsa Hosseini Nayeri - Iran

In the figure below points $A,B$ are the centers of the circles $\omega_1, \omega_2$. Starting from the line $BC$ points $E,F,G,H,I$ are obtained respectively. Find the angle $\angle IBE$.

Points $X,Y$ lie on the side $CD$ of a convex pentagon $ABCDE$ with $X$ between $Y$ and $C$. Suppose that the triangles $\bigtriangleup XCB, \bigtriangleup ABX, \bigtriangleup AXY, \bigtriangleup AYE, \bigtriangleup YED$ are all similar (in this exact order). Prove that circumcircles of the triangles $\bigtriangleup ACD, \bigtriangleup AXY$ are tangent. Pouria Mahmoudkhan Shirazi - Iran

Let $\bigtriangleup ABC$ be an acute triangle with a point $D$ on side $BC$. Let $J$ be a point on side $AC$ such that $\angle BAD = 2\angle ADJ$, and $\omega$ be the circumcircle of triangle $\bigtriangleup CDJ$. The line $AD$ intersects $\omega$ again at a point $P$, and $Q$ is the feet of the altitude from $J$ to $AB$. Prove that if $JP = JQ$, then the line perpendicular to $DJ$ through $A$ is tangent to $\omega$. Proposed by Ivan Chan - Malaysia

Eric has assembled a convex polygon $P$ from finitely many centrally symmetric (not necessarily congruent or convex) polygonal tiles. Prove that $P$ is centrally symmetric. Proposed by Josef Tkadlec - Czech Republic

Point $P$ is the intersection of diagonals $AC,BD$ of the trapezoid $ABCD$ with $AB \parallel CD$. Reflections of the lines $AD$ and $BC$ into the internal angle bisectors of $\angle PDC$ and $\angle PCD$ intersects the circumcircles of $\bigtriangleup APD$ and $\bigtriangleup BPC$ at $D'$ and $C'$. Line $C'A$ intersects the circumcircle of $\bigtriangleup BPC$ again at $Y$ and $D'C$ intersects the circumcricle of $\bigtriangleup APD$ again at $X$. Prove that $P,X,Y$ are collinear. Proposed by Iman Maghsoudi - Iran

An equilateral triangle $\bigtriangleup ABC$ is split into $4$ triangles with equal area; three congruent triangles $\bigtriangleup ABX,\bigtriangleup BCY, \bigtriangleup CAZ$, and a smaller equilateral triangle $\bigtriangleup XYZ$, as shown. Prove that the points $X, Y, Z$ lie on the incircle of triangle $\bigtriangleup ABC$. Proposed by Josef Tkadlec - Czech Republic

Point $P$ lies on the side $CD$ of the cyclic quadrilateral $ABCD$ such that $\angle CBP = 90^{\circ}$. Let $K$ be the intersection of $AC,BP$ such that $AK = AP = AD$. $H$ is the projection of $B$ onto the line $AC$. Prove that $\angle APH = 90^{\circ}$. Proposed by Iman Maghsoudi - Iran

In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$. Proposed Michal Jan'ik - Czech Republic

Point $P$ is inside the acute triangle $\bigtriangleup ABC$ such that $\angle BPC=90^{\circ}$ and $\angle BAP=\angle PAC$. Let $D$ be the projection of $P$ onto the side $BC$. Let $M$ and $N$ be the incenters of the triangles $\bigtriangleup ABD$ and $\bigtriangleup ADC$ respectively. Prove that the quadrilateral $BMNC$ is cyclic. Proposed by Hussein Khayou - Syria

Cyclic quadrilateral $ABCD$ with circumcircle $\omega$ is given. Let $E$ be a fixed point on segment $AC$. $M$ is an arbitrary point on $\omega$, lines $AM$ and $BD$ meet at a point $P$. $EP$ meets $AB$ and $AD$ at points $R$ and $Q$, respectively, $S$ is the intersection of $BQ,DR$ and lines $MS$ and $AC$ meet at a point $T$. Prove that as $M$ varies the circumcircle of triangle $\bigtriangleup CMT$ passes through a fixed point other than $C$. Proposed by Chunlai Jin - China