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

2025 Bangladesh Mathematical Olympiad, P9
Tags: geometry , altitude , similarity , parallel lines , congruent triangles
Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, respectively. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ and $N$, respectively, other than $D$. Prove that $BN = LC$.

1978 Poland - Second Round, 2
Tags: parallel lines , intersection , geometry
In the plane, a set of points $ M $ is given with the following properties: 1. The points of the set $ M $ do not lie on one straight line, 2. If the points $ A, B, C$, and $D$ are vertices of a parallelogram and $ A, B, C \in M $, then $ D \in M $, 3. If $ A, B \in M $, then $ AB \geq 1 $. Prove that there exist two families of parallel lines such that $ M $ is the set of all intersection points of the lines of the first family with the lines of the second family.

Novosibirsk Oral Geo Oly VIII, 2023.7
Tags: geometry , square , parallel lines
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]