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.

AND:
OR:
NO:

Found problems: 28

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]

2018 Hanoi Open Mathematics Competitions, 3
Tags: combinatorial geometry , combinatorics , parallel lines
The lines $\ell_1$ and \ell_2 are parallel. The points $A_1,A_2, ...,A_7$ are on $\ell_1$ and the points $B_1,B_2,...,B_8$ are on $\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure). The [b]greatest number[/b] of intersection points is [img]https://cdn.artofproblemsolving.com/attachments/4/9/92153dce5a48fcba0f5175d67e0750b7980e84.png[/img] A. $580$ B. $585$ C. $588$ D. $590$ E. $593$

2021 Lotfi Zadeh Olympiad, 1
Tags: geometry , parallel lines , cyclic quadrilateral
In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.