MathDB

2

Part of 2015 China National Olympiad

Problems(2)

Graph with no K_5

Source: China Mathematical Olympiad 2015 Q5

12/21/2014
Given 3030 students such that each student has at most 55 friends and for every 55 students there is a pair of students that are not friends, determine the maximum kk such that for all such possible configurations, there exists kk students who are all not friends.
combinatorics proposedcombinatorics
Two ratio are equal

Source: 2015 China National Olympiad

12/20/2014
Let A,B,D,E,F,C A, B, D, E, F, C be six points lie on a circle (in order) satisfy AB=AC AB=AC . Let P=AD∩BE,R=AF∩CE,Q=BF∩CD,S=AD∩BF,T=AF∩CD P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD . Let K K be a point lie on ST ST satisfy ∠QKS=∠ECA \angle QKS=\angle ECA .
Prove that SKKT=PQQR \frac{SK}{KT}=\frac{PQ}{QR}
ratiotrigonometrygeometry