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: 3

2024 Miklos Schweitzer, 8
Tags: graph theory , planar graph , combinatorial geometry
Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.

2016 JBMO Shortlist, 4
Tags: combinatorics , planar graph , triangulation , graph theory
A splitting of a planar polygon is a fi nite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

KoMaL A Problems 2019/2020, A. 777
Tags: combinatorics , graph theory , planar graph
A finite graph $G(V,E)$ on $n$ points is drawn in the plane. For an edge $e$ of the graph, let $\chi(e)$ denote the number of edges that cross over edge $e$. Prove that \[\sum_{e\in E}\frac{1}{\chi(e)+1}\leq 3n-6.\][i]Proposed by Dömötör Pálvölgyi, Budapest[/i]