Olympiad problems

2022 Switzerland Team Selection Test, 2

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

2024 Euler Olympiad, Round 2, 3

Tags: euler , quadrilateral , geometry

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]

1994 Chile National Olympiad, 6

Tags: geometry , parallelogram , quadrilateral , projection

On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.

2025 AIME, 14

Tags: geometry , quadrilateral

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 $n.$

2007 Sharygin Geometry Olympiad, 2

Tags: diagonal , geometry , quadrilateral , isosceles , isosceles triangle

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

2022 SAFEST Olympiad, 5

Tags: tangency , collinear , quadrilateral , geometry

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

2008 Germany Team Selection Test, 2

Tags: geometry , quadrilateral , incircle , triangle

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]