MathDB

2

Part of 2001 Hungary-Israel Binational

Problems(2)

Find the locus

Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

4/12/2007
Points A,B,C,DA, B, C, D lie on a line ll, in that order. Find the locus of points PP in the plane for which ∠APB=∠CPD\angle{APB}=\angle{CPD}.
geometry proposedgeometry
$G_n$ contains two triangles that share exactly one vertex

Source: 12-th Hungary-Israel Mathematical Competition 2001

4/16/2007
Here GnG_{n} denotes a simple undirected graph with nn vertices, KnK_{n} denotes the complete graph with nn vertices, Kn,mK_{n,m} the complete bipartite graph whose components have mm and nn vertices, and CnC_{n} a circuit with nn vertices. The number of edges in the graph GnG_{n} is denoted e(Gn)e(G_{n}). If n≥5n \geq 5 and e(Gn)≥n24+2e(G_{n}) \geq \frac{n^{2}}{4}+2, prove that GnG_{n} contains two triangles that share exactly one vertex.
graph theorycombinatorics proposedcombinatorics