3

Part of 2008 International Zhautykov Olympiad

Problems(2)

Hard combinatorics problem

Source: Problem 3 from ZIMO 2008

Let A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N} , 1\leq a_i\leq i \plus{} 1 for each i \equal{} 1,2\dots,8\}.A subset

X\subset A

is called sparse if for each two distinct elements

(a_1,\dots,a_8)

(b_1,\dots,b_8)\in X

,there exist at least three indices

i

a_i\neq b_i

. Find the maximal possible number of elements in a sparse subset of set

A

combinatorics proposedcombinatorics

Three variables, cyclic

Source: Zhautykov Olympiad, Kazakhstan 2008, Question 6

a, b, c

be positive integers for which abc \equal{} 1. Prove that \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}.

inequalitiesinvariantthree variable inequality