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Eventually double of the previous term

Source: Own. Malaysian APMO CST 2024 P1

February 24, 2024
number theory

Problem Statement

Let a1<a2<⋯a_1<a_2< \cdots be a strictly increasing sequence of positive integers. Suppose there exist NN such that for all n>Nn>N, an+1∣a1+a2+⋯+ana_{n+1}\mid a_1+a_2+\cdots+a_n Prove that there exist MM such that am+1=2ama_{m+1}=2a_m for all m>Mm>M.
Proposed by Ivan Chan Kai Chin
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