Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.

$ABCD$ is a cyclic quadrilateral. The lines $AD$ and $BC$ meet at X, and the lines $AB$ and $CD$ meet at $Y$ . The line joining the midpoints $M$ and $N$ of the diagonals $AC$ and $BD$, respectively, meets the internal bisector of angle $AXB$ at $P$ and the external bisector of angle $BYC$ at $Q$. Prove that $PXQY$ is a rectangle

In an inscribed quadrilateral $ABCD$ we have $BC = CD$. Prove that the area of the quadrilateral is equal to $\frac{(AC)^2 \sin A}{2}$ (D. Fomin, Leningrad)

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Let $A_1B_3A_2B_1A_3B_2$ be a cyclic hexagon such that $A_1B_1,A_2B_2,A_3B_3$ intersect at one point. Let $C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2$. Let $D_1$ be the point on the circumcircle of the hexagon such that $C_1B_1D_1$ touches $A_2A_3$. Define $D_2,D_3$ analogously. Show that $A_1D_1,A_2D_2,A_3D_3$ meet at one point.

The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$. The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$, and $F$ is the midpoint of $TN$. Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$, respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$, and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$, respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.

Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear. .

Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

Determine all real tuples $\left(x_1,x_2,x_3,x_4,x_5,x_6\right)$ such that \begin{align*} x_1(x_6 + x_2) &= x_3 + x_5, \\ x_2(x_1 + x_3) &= x_4 + x_6, \\ x_3(x_2 + x_4) &= x_5 + x_1, \\ x_4(x_3 + x_5) &= x_6 + x_2, \\ x_5(x_4 + x_6) &= x_1 + x_3, \\ x_6(x_5 + x_1) &= x_2 + x_4. \end{align*}

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?

The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$. The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$. (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.

In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$, the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$. (a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle. (b) Find the area of quadrilateral $ABCD$.

Given a circle with radius $R$ and inscribed $n$-gon with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.

In $\triangle ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD = EC$. Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$, and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$. Prove that $D$, $E$, $X$, and $Y$ are concyclic.

Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with center $O$. Let $F$ be the second intersection point of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the intersection point of the segments $AC$ and $BD$. (A Zaslavskiy)

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic