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Arithmetic Sequence of Products

Source: IMO Shortlist 2023 N4

July 17, 2024
arithmetic sequencenumber theory

Problem Statement

Let a1,…,an,b1,…,bna_1, \dots, a_n, b_1, \dots, b_n be 2n2n positive integers such that the n+1n+1 products a1a2a3⋯an,b1a2a3⋯an,b1b2a3⋯an,…,b1b2b3⋯bna_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
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