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a_ia_{i+1}a_{i+2}a_{i+3}=i(mod p)

Source: IMO SL 2020 N1

July 20, 2021
number theorySequenceDivisibilityIMO ShortlistIMO Shortlist 2020

Problem Statement

Given a positive integer kk show that there exists a prime pp such that one can choose distinct integers a1,a2⋯ ,ak+3∈{1,2,⋯ ,p−1}a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\} such that p divides aiai+1ai+2ai+3−ia_ia_{i+1}a_{i+2}a_{i+3}-i for all i=1,2,⋯ ,ki= 1, 2, \cdots, k.
South Africa
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