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Existence of $a_1,a_2,\dots,a_{8002}$ such that $a_ia_ja_k|P(a_i)P(a_j)P(a_k)$

Source: St. Petersburg MO 2000, 11th grade, P4

April 22, 2023

Polynomialsinteger polynomialsnumber theoryExistence

Problem Statement

Let

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

. Prove that there don't exist 8002 distinct positive integers

a_1,\dots,a_{8002}

such that

a_ia_ja_k|P(a_i)P(a_j)P(a_k)

for all

i\neq j\neq k

.
[I]Proposed by A. Baranov