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IberoAmerican
2023 Iberoamerican
6
6
Part of
2023 Iberoamerican
Problems
(1)
P-representable Polynomial
Source: IberoAmerican, Day 2, P6
9/9/2023
Let
P
P
P
be a polynomial of degree greater than or equal to
4
4
4
with integer coefficients. An integer
x
x
x
is called
P
P
P
-representable if there exists integer numbers
a
a
a
and
b
b
b
such that
x
=
P
(
a
)
−
P
(
b
)
x = P(a) - P(b)
x
=
P
(
a
)
−
P
(
b
)
. Prove that, if for all
N
≥
0
N \geq 0
N
≥
0
, more than half of the integers of the set
{
0
,
1
,
…
,
N
}
\{0,1,\dots,N\}
{
0
,
1
,
…
,
N
}
are
P
P
P
-representable, then all the even integers are
P
P
P
-representable or all the odd integers are
P
P
P
-representable.
algebra
polynomial