There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that \[ \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} \] if and only if the diagonals $ AC$ and $ BD$ are perpendicular. [i]Proposed by Dusan Djukic, Serbia[/i]

It is known that the bisector of the angle $\angle ADC$ of the inscribed quadrilateral $ABCD$ passes through the center of the circle inscribed in the triangle $ABC$. Let $M$ be an arbitrary point of the arc $ABC$ of the circle circumscribed around $ABCD$. Denote by $P$ and $Q$ the centers of the circles inscribed in the triangles $ABM$ and $BCM$. Prove that all triangles $DPQ$ obtained by moving point $M$ are similar to each other. Find the angle $\angle PDQ$ and ratio $BP : PQ$ if $\angle BAC = \alpha$, $\angle BCA = \beta$

Let $\Gamma, \Gamma_1, \Gamma_2$ be mutually tangent circles. The three circles are also tangent to a line $l$. Let $\Gamma, \Gamma_1$ be tangent to each other at $B_1$, $\Gamma, \Gamma_2$ be tangent to each other at $B_2$, $\Gamma_1, \Gamma_2$ be tangent to each other at $C$. $\Gamma, \Gamma_1, \Gamma_2$ are tangent to $l$ at $A, A_1, A_2$ respectively, where $A$ is between $A_1,A_2$. Let $D_1 = A_1C \cap A_2B_2, D_2 = A_2C \cap A_1B_1$. Prove that $D_1D_2$ is parallel to $l$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$ intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$ and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets $AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line through $S$ parallel to $PQ$ is tangent to $\Omega$.

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

Let $BE$ and $CF$ be the altitudes of an acute triangle $ABC$ with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively. such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$. Prove that $FM$ is parallel to $EN$.

In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4:5$. What is the degree measure of angle $BCD$? [asy] unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair A=(-3cos(80),-3sin(80)); pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); draw(E--A); draw(B--A); draw(E--D); draw(C--D); draw(B--C); pair[] ps={A,B,C,D,E,O}; dot(ps); label("$A$",A,N); label("$B$",B,NE); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$$",O,N);[/asy] $ \textbf{(A)}\ 120 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 130 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 140 $

Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).

On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$