BD TST 2021 NT Exam P1, IMO SL 2021 N1 - Show that $p\mid a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i$

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Anindya Biswas
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BD TST 2021 NT Exam P1, IMO SL 2021 N1 - Show that $p\mid a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i$

Unread post by Anindya Biswas » Mon Aug 02, 2021 4:10 pm

Given a positive integer $k$, show that there exists a prime $p$ such that one can choose distinct integers $a_1, a_2, \dots , a_{k+3} \in \{1, 2, \dots, p-1\}$ such that $p$ divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i = 1, 2,\dots, k$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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