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Moldova Contests
Moldova Team Selection Test
1997 Moldova Team Selection Test
11
11
Part of
1997 Moldova Team Selection Test
Problems
(1)
Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$
Source: Moldova TST 1997
8/8/2023
Let
P
(
X
)
P(X)
P
(
X
)
be a polynomial with real coefficients such that
{
P
(
n
)
}
≤
1
n
,
∀
n
∈
N
\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}
{
P
(
n
)}
≤
n
1
,
∀
n
∈
N
, where
{
a
}
\{a\}
{
a
}
is the fractional part of the number
a
a
a
. Show that
P
(
n
)
∈
Z
,
∀
n
∈
N
P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}
P
(
n
)
∈
Z
,
∀
n
∈
N
.
algebra
polynomial