3

Part of 1993 Cono Sur Olympiad

Problems(2)

Source: Cono Sur 1993-problem 3

Find the number of elements that a set

B

can have, contained in

(1, 2, ... , n)

, according to the following property: For any elements

a

b

B

a \ne b

(a-b) \not| (a+b)

ceiling functionnumber theory unsolvednumber theory

Proof with points

Source: Cono Sur 1993-problem 6

Prove that, given a positive integrer

n

, there exists a positive integrer

k_n

with the following property: Given any

k_n

points in the space,

4

4

non-coplanar, and associated integrer numbers between

1

n

to each sharp edge that meets

2

of this points, there's necessairly a triangle determined by

3

of them, whose sharp edges have associated the same number.

combinatorics unsolvedcombinatorics