MathDB

3

Part of 2013 Turkey MO (2nd round)

Problems(2)

Simple connected graph with 100 vertices and 2013 edges

Source: Turkey National Olympiad Second Round 2013 P3

11/28/2013
Let GG be a simple, undirected, connected graph with 100100 vertices and 20132013 edges. It is given that there exist two vertices AA and BB such that it is not possible to reach AA from BB using one or two edges. We color all edges using nn colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of nn.
graph theorycombinatoricsGraph coloringConnected graphs
n points with integer distances

Source: Turkey National Olympiad Second Round 2013 P6

11/28/2013
Let nn be a positive integer and P1,P2,…,PnP_1, P_2, \ldots, P_n be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances PiP1,PiP2,…,PiPnP_iP_1, P_iP_2, \ldots, P_iP_n forms the same sequence for all i=1,2,…,ni=1,2, \ldots, n when these numbers are arranged in a non-decreasing order. Find all possible values of nn.
geometryrectanglegeometry proposed