Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively. Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\). [i]Proposed by Zaza Meliqidze, Georgia [/i]

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

The point $M$ is inside the convex quadrilateral $ABCD$, such that \[ MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD} \quad \textnormal{and} \quad \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. \] Prove that $AB \cdot CM = BC \cdot MD$ and $BM \cdot AD = MA \cdot CD.$

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds \[S \leq \frac{1}{2}(ac+bd).\] For which quadrangles does the inequality become equality?

Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

In a convex quadrilateral $ABCD$, $\angle BAC=\angle DAC=55^\circ$, $\angle DCA=20^\circ$, and $\angle BCA=15^\circ$. Find the measure of $\angle DBA$.

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

Let $ABCD$ be a quadrilateral, such that $AB = BC = CD.$ There are points $X, Y$ on rays $CA, BD,$ respectively, such that $BX = CY.$ Let $P, Q, R, S$ be the midpoints of segments $BX, CY ,$ $XD, YA,$ respectively. Prove that points $P, Q, R, S$ lie on a circle.

Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.

The diagonals of a quadrilateral $ ABCD$ are perpendicular: $ AC \perp BD.$ Four squares, $ ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $ CL, DF, AH, BJ$ are denoted by $ P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $ AI, BK, CE DG$ are denoted by $ P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $ P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $ P_1,Q_1,R_1, S_1$ and $ P_2,Q_2,R_2, S_2$ are the two quadrilaterals. [i]Alternative formulation:[/i] Outside a convex quadrilateral $ ABCD$ with perpendicular diagonals, four squares $ AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $ Q_1$ and $ Q_2$ formed by the lines $ AG, BI, CK, DE$ and $ AJ, BL, CF, DH,$ respectively, are congruent.

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

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]

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.