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Sum of integers of the same colour has the same colour again

Source: Germany 2014- Problem 3

December 4, 2022

combinatoricscolouringSumAdditive combinatorics

Problem Statement

Given two positive integers

n

and

k

, we say that

k

n

-ergetic if: However the elements of

M=\{1,2,\ldots, k\}

are coloured in red and green, there exist

n

not necessarily distinct integers of the same colour whose sum is again an element of

M

of the same colour. For each positive integer

n

, determine the least

n

-ergetic integer, if it exists.