2

Part of 2008 China National Olympiad

Problems(2)

partition into subsets not containing convex sequences

Source:

Given an integer

n\ge3

, prove that the set

X=\{1,2,3,\ldots,n^2-n\}

can be divided into two non-intersecting subsets such that neither of them contains

n

a_1,a_2,\ldots,a_n

a_1<a_2<\ldots<a_n

a_k\le\frac{a_{k-1}+a_{k+1}}2

k=2,\ldots,n-1

inductioninequalitiescombinatorics proposedcombinatorics

China Mathematics Olympiads (National Round) 2008 Problem 5

Source:

Find the smallest integer

n

satisfying the following condition: regardless of how one colour the vertices of a regular

n

-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.

geometrytrapezoidpigeonhole principlecombinatorics