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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2000 Saint Petersburg Mathematical Olympiad
11.4
11.4
Part of
2000 Saint Petersburg Mathematical Olympiad
Problems
(1)
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
4/22/2023
Let
P
(
x
)
=
x
2000
−
x
1000
+
1
P(x)=x^{2000}-x^{1000}+1
P
(
x
)
=
x
2000
−
x
1000
+
1
. Prove that there don't exist 8002 distinct positive integers
a
1
,
…
,
a
8002
a_1,\dots,a_{8002}
a
1
,
…
,
a
8002
such that
a
i
a
j
a
k
∣
P
(
a
i
)
P
(
a
j
)
P
(
a
k
)
a_ia_ja_k|P(a_i)P(a_j)P(a_k)
a
i
a
j
a
k
∣
P
(
a
i
)
P
(
a
j
)
P
(
a
k
)
for all
i
≠
j
≠
k
i\neq j\neq k
i
=
j
=
k
.[I]Proposed by A. Baranov
Polynomials
integer polynomials
number theory
Existence