MathDB

3

Part of 1993 China Team Selection Test

Problems(2)

n-colored graph without triangles

Source: China TST 1993, problem 3

6/27/2005
A graph G=(V,E)G=(V,E) is given. If at least nn colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''n−n-colored''. Prove that for any n∈Nn \in \mathbb{N}, there is a n−n-colored graph without triangles.
graph theorycombinatorics unsolvedcombinatorics
Extend DP until it cuts the circumcircle again at N

Source: China TST 1993, problem 6

6/27/2005
Let ABCABC be a triangle and its bisector at AA cuts its circumcircle at D.D. Let II be the incenter of triangle ABC,ABC, MM be the midpoint of BC,BC, PP is the symmetric to II with respect to MM (Assuming PP is in the circumcircle). Extend DPDP until it cuts the circumcircle again at N.N. Prove that among segments AN,BN,CNAN, BN, CN, there is a segment that is the sum of the other two.
geometrycircumcircleincentertrigonometrygeometric transformationreflectionRing Theory