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Maximising a sequence-given value

Source: IMO Shortlist 2018 A4

July 17, 2019

algebraSequencesIMO Shortlistmaximum valueIMO shortlist 2018

Problem Statement

Let

a_0,a_1,a_2,\dots

be a sequence of real numbers such that

a_0=0, a_1=1,

and for every

n\geq 2

there exists

1 \leq k \leq n

satisfying

a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}.

Find the maximum possible value of

a_{2018}-a_{2017}