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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2000 Saint Petersburg Mathematical Olympiad
9.3
9.3
Part of
2000 Saint Petersburg Mathematical Olympiad
Problems
(1)
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
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
. Do there exist distinct positive integers
a
1
,
…
,
a
2001
a_1,\dots,a_{2001}
a
1
,
…
,
a
2001
such that
a
i
a
j
∣
P
(
a
i
)
P
(
a
j
)
a_ia_j|P(a_i)P(a_j)
a
i
a
j
∣
P
(
a
i
)
P
(
a
j
)
for all
i
≠
j
i\neq j
i
=
j
?[I]Proposed by A. Baranov
number theory
Polynomials
integer polynomials
Existence
St. Petersburg MO