Olympiad problems

2011 IFYM, Sozopol, 1

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let the lines $AC\cap BD=O$, $AD\cap BC=P$, and $AB\cap CD=Q$. Line $QO$ intersects $k$ in points $M$ and $N$. Prove that $PM$ and $PN$ are tangent to $k$.

1975 Bundeswettbewerb Mathematik, 3

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

quadrilateral circumscribed quadrilateral geometry inscribed diagonal quadrilaterals

2019 Oral Moscow Geometry Olympiad, 2

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

quadrilateral geometry congruent diagonal congruence modular arithmetic quadrilaterals

2020 AMC 12/AHSME, 18

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$? $\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$

area quadrilateral geometry Pythagorean theorem law of cosines quadrilaterals similar triangles

2011 Kyiv Mathematical Festival, 3

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?

cut quadrilateral geometry quadrilaterals

1967 IMO Shortlist, 1

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.

area maximization quadrilateral geometry quadrilaterals

2009 Germany Team Selection Test, 2

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]

trigonometry homothety circumcircle quadrilateral geometry inversion angle chasing cyclic quadrilaterals geometric inversion law of sines power of a point quadrilaterals radical axis