3

Part of 2009 China National Olympiad

Problems(2)

The number of convex m polygon

Source: Chinese National Olympiad 2009 P3

Given two integers

m,n

4 < m < n.

Let A_{1}A_{2}\cdots A_{2n \plus{} 1} be a regular 2n\plus{}1 polygon. Denote by

P

the set of its vertices. Find the number of convex

m

polygon whose vertices belongs to

P

and exactly has two acute angles.

inductioncombinatorics proposedcombinatorics

Source: Chinese National Olympiad 2009 P6

Given an integer

n > 3.

Prove that there exists a set

S

n

pairwisely distinct positive integers such that for any two different non-empty subset of

S

A,B, \frac {\sum_{x\in A}x}{|A|}

\frac {\sum_{x\in B}x}{|B|}

are two composites which share no common divisors.

inductionnumber theorygreatest common divisorcombinatorics proposedcombinatorics