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Sets of numbers called balanced sets

Source: Baltic Way 2015

November 8, 2015

number theory

Problem Statement

A subset

S

of

{1,2,...,n}

is called balanced if for every

a

from

S

there exists some

b

from

S

,

b\neq a

, such that

\frac{(a+b)}{2}

is in

S

as well. (a) Let

k > 1

be an integer and let

n = 2k

. Show that every subset

S

of

{1,2,...,n}

with

|S| > \frac{3n}{4}

is balanced. (b) Does there exist an

n =2k

, with

k > 1

an integer, for which every subset

S

of

{1,2,...,n}

with

|S| >\frac{2n}{3}

is balanced?

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