4

Part of 2015 China Northern MO

Problems(2)

partition of 1-16 with a subset with elements a,b,c such that a+b=c

Source: China Northern MO 2015 grade 10 p4 CNMO

S = \{1,2,3,…,16\}

is partitioned into

n

subsets, there must be a subset in which elements

a, b, c

(can be the same) exist, satisfying

a+ b=c

. Find the maximum value of

n

number theorycombinatorics

combo NT with 108 pos. integers <= 2015

Source: China Northern MO 2015 grade 11 p4 CNMO

It is known that

a_1, a_2,...a_{108}

108

different positive integers not exceeding

2015

. Prove that there is a positive integer

k

such that there are at least four different pairs

(i, j)

a_i-a_j =k

combinatoricsnumber theory