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Sequence a_(n+1)=a_n

Source: Shortlist BMO 2019, A1

November 7, 2020
algebranumber theory

Problem Statement

Let a0a_0 be an arbitrary positive integer. Consider the infinite sequence (an)n≥1(a_n)_{n\geq 1}, defined inductively as follows: given a0,a1,...,an−1a_0, a_1, ..., a_{n-1} define the term ana_n as the smallest positive integer such that a0+a1+...+ana_0+a_1+...+a_n is divisible by nn. Prove that there exist a positive integer a positive integer MM such that an+1=ana_{n+1}=a_n for all n≥Mn\geq M.
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