MathDB

Let

S

be a set of 2006 numbers. We call a subset

T

S

naughty if for any two arbitrary numbers

u

v

(not neccesary distinct) in

T

u+v

is not in

T

. Prove that 1) If

S=\{1,2,\ldots,2006\}

every naughty subset of

S

has at most 1003 elements; 2) If

S

is a set of 2006 arbitrary positive integers, there exists a naughty subset of

S

which has 669 elements.

Problem Statement