This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 28

2023 Sharygin Geometry Olympiad, 20
Tags: geometry , Isogonal conjugate , tangent , symmedian , Parallel Lines , Sharygin Geometry Olympiad , Sharygin 2023
Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.

1999 Spain Mathematical Olympiad, 6
Tags: combinatorial geometry , Parallel Lines , minimum , Plane , combinatorics
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

2022 JBMO TST - Turkey, 4
Tags: geometry , Parallel Lines , Angle condition
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that $$m(\widehat{ACB})=m(\widehat{PCT})$$