6

Part of 1991 Czech And Slovak Olympiad IIIA

Problems(1)

partition of N in 3 subsets, a_1 < a_2 < ... < a_k and a_{j+1} -a_j \le m

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1991 p6

N

is partitioned into three subsets

A_1,A_2,A_3

. Prove that at least one of them has the following property: There exists a positive number

m

such that for any

k

one can find numbers

a_1 < a_2 < ... < a_k

in that subset satisfying

a_{j+1} -a_j \le m

j = 1,...,k -1

partitionSubsetscombinatorics