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Hong Kong Contests
Hong Kong National Olympiad
2001 Hong kong National Olympiad
3
$P(x)\in\mathbb{Z}[x]$
$P(x)\in\mathbb{Z}[x]$
Source: 4-th Hong Kong Mathematical Olympiad 2001
February 15, 2007
algebra proposed
algebra
Problem Statement
Let
k
≥
4
k\geq 4
k
≥
4
be an integer number.
P
(
x
)
∈
Z
[
x
]
P(x)\in\mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
such that
0
≤
P
(
c
)
≤
k
0\leq P(c)\leq k
0
≤
P
(
c
)
≤
k
for all
c
=
0
,
1
,
.
.
.
,
k
+
1
c=0,1,...,k+1
c
=
0
,
1
,
...
,
k
+
1
. Prove that
P
(
0
)
=
P
(
1
)
=
.
.
.
=
P
(
k
+
1
)
P(0)=P(1)=...=P(k+1)
P
(
0
)
=
P
(
1
)
=
...
=
P
(
k
+
1
)
.
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