4

Part of 2012 Romania Team Selection Test

Problems(2)

Atoms among 100 digit numbers

Source: Romania TST 3 2012, Problem 4

S

be a set of positive integers, each of them having exactly

100

10

representation. An element of

S

is called atom if it is not divisible by the sum of any two (not necessarily distinct) elements of

S

S

contains at most

10

atoms, at most how many elements can

S

inductionvectorcombinatorics proposedcombinatorics

Special orientation of planar graph

Source: Romania TST 1 2012, Problem 4

Prove that a finite simple planar graph has an orientation so that every vertex has out-degree at most 3.

inductioncombinatorics proposedcombinatorics