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Existence of $a_1,\dots,a_{2001}$ such that $a_ia_j|P(a_i)P(a_j)$

Source: St. Petersburg MO 2000, 9th grade, P3

April 22, 2023

number theoryPolynomialsinteger polynomialsExistenceSt. Petersburg MO

Problem Statement

Let

P(x)=x^{2000}-x^{1000}+1

. Do there exist distinct positive integers

a_1,\dots,a_{2001}

such that

a_ia_j|P(a_i)P(a_j)

for all

i\neq j

?
[I]Proposed by A. Baranov