Problem

Source: IMO Shortlist 2013, Combinatorics #5

Tags: function, combinatorics, Additive combinatorics, Sequence, IMO Shortlist



Let r be a positive integer, and let a0,a1, be an infinite sequence of real numbers. Assume that for all nonnegative integers m and s there exists a positive integer n[m+1,m+r] such that am+am+1++am+s=an+an+1++an+s Prove that the sequence is periodic, i.e. there exists some p1 such that an+p=an for all n0.