10

Part of 2015 Baltic Way

Problems(1)

Sets of numbers called balanced sets

Source: Baltic Way 2015

S

{1,2,...,n}

is called balanced if for every

a

S

there exists some

b

S

b\neq a

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

S

as well. (a) Let

k > 1

be an integer and let

n = 2k

. Show that every subset

S

{1,2,...,n}

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

is balanced. (b) Does there exist an

n =2k

k > 1

an integer, for which every subset

S

{1,2,...,n}

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